is telling your employer you may leave if passive aggressive coworkers don't stop gossiping about you behind your back and gaslighting you a good idea?

Unless they need you more than you need them, announcing your intent to leave "unless..." puts you in a more difficult negotiating position because you have signaled you have other options that, perhaps more challenging for you, are an easier solution than them having to fix it for you. HR isn't your friend.
Practically and more directly, if you can be easily replaced then your workplace issues are yours. If you are difficult or expensive to replace, your workplace issues are theirs.
- Take it from experience @vestmoria@linux.community this advice is true. Going to HR means that they're going to take action on the one causing the most noise, and when you go to them at that moment, that's you. That may sound discouraging, but that's what it is. Something like gossip is something HR will be willing to ignore. By going to them you are creating the most noise so it will only adversely affect you.
  It doesn't matter if you're right. It doesn't matter if you're a great employee. You are officially making negative noise, that's a liability for the company. They'll smooth you over, tell you there's a plan, but it's officially on record that you'll speak up while others quietly work, and in the corporate world that's a black mark.
  Take it from me, just silently start looking for new work. Right now you have all the time in the world, search around, be picky, get something better. Leave them behind you. That's the only real way to propel your career forward.
  
  Depends. HR maybe see your complaint and decide "I should take care of this workplace harassment before someone wisens up and talks to a lawyer"
  I would say talk to HR, but not until you've actually got another job lined up.
  Maybe they think ahead and take care of the problem, maybe they just sweep shit under the rug. It depends on the HR rep in question.
  
  but it’s officially on record that you’ll speak up while others quietly work
  why would I want to work where people try destroying my credibility behind my back? This is not something I'm willing to overlook.
  If HR acts like you described, if I'm that replaceable to them, so is my workplace.
  The others don't work quietly, btw.
  ETA: wait, are you implying this is normalized? Employees do actually say nothing not to land in hot water, because they're afraid of being fired and are willing to overlook the gossip and backstabbing for a check? Not for me.

I would tell a manager or HR about your problems but not tell them you plan to leave unless you are ready to get fired or hours cut.

Like others have said (or hinted at), never make an ultimatum you are unwilling or unable to follow through on.
And saying you'll leave "unless x happens" is just such an ultimatum.

HR exists to protect the company. When you raise a problem to them, you are the one responsible for the problem being put on record. This means you are the problem. Even if you go to HR and they do something about it, you still lose because your coworkers will steer clear of you for being a narc and any chance at a career is pretty much gone. Either work the problem out with your coworkers yourself, ignore it and keep your head down, or find a new job.
- I agree with the HR department but calling someone a narc for raising an issue of bullying is called victimblaming.
  I‘ve pulled tons of money out of companies for being bullied since they are responsible for keeping the workplace secure. Being a minority and bullied for it is great reason to sue and should be done in that case.
  This advice is essentially „be a good little *** and take it in stride“. I cant believe that its upvoted so heavily.
  
  Woah, dial it it back a bit. We're not talking about racial discrimination or violence. OP said people were talking behind their backs and provided no additional details. If you file a HR complaint just because you saw someone whispering while looking at you, you're 100% a narc. If you file a racial discrimination lawsuit because you're a minority and the company isn't firing people you don't like, you're the worst kind of person. If there's more to the story, OP didn't share it.
- in addition to you being an asshole, you're also wrong in practice about how HR teams work. If they hear about shit like this, they really do try to do something about it. Sometimes they can't really accomplish anything, and they're just bureaucratic about it, but no, they do not think of the person making the report as a problem, they think about the person actually causing the actual problem. Hostile work environments are unproductive, are bad for employee retention, and have a heightened risk of law suit. Only shitty businesspeople think the problem doesn't exist as long as it's not on paper.
  OP's better off if their employees steer clear of them—that much is obvious, isn't it?

As long as you're willing and able to follow through with it if needed.
A better approach might be to ask them to fix it without the ultimatum first

A hostile work environment is grounds for a lawsuit. Document everything, bring it to a lawyer for council. If you have a case, they will tell you whether to talk to HR. If you don't have a case, you have no leverage here.

If you have problems with your team members you should being it up with your manager. But don't just tell them the problem and expect them to come up with a solution.
Documentation on when events happen and how you believe a change in attitude will make the org a better place to work will get a far more favourable response.
Now if you talk with your manager and nothing changes, then look for a new job and on your HR exit interview explicitly say why you are leaving.
- They need to find a witness. Otherwise it’s a he said, she said issue that can’t be confirmed as harassment.
- HR takes a different tact when a person starts to sight harassment from contemporaneous notes.

I have only ever worked in germany so thats all I can talk about from a legal perspective (not a lawyer):
If people bully you and you can find evidence for it, the colleagues, maybe your manager and the whole company can be shit out of luck and you win. A lot of folks have successfully sued (which is very cheap in germany, you dont even need a lawyer at first) for „mobbing“ (bullying essentially).
If one or more colleagues bully you, you collect evidence, confront your manager. They then have reason to terminate the offender if the evidence is strong.
If the manager doesnt act, you have reason to sue and get considerable sums out of the company.
But yes, if people go behind your back and oust you, they are in the wrong. Even if you did wrong, they should talk to you about it or involve management. Social ousting is never an acceptable reaction imo.
- In the states, you'd probably be let go for something like this. At will employment means you can be let go for any or no reason, it's total BS. Google just recently waited until a group of union members were in front of city council to fire them.
  
  At will employment is modern slavery imo.

If HR and manager do nothing to address the problem, wouldn’t it be a better strategy to start working the least possible and let the company fire me, while looking for another job?
Doing your job and looking for another job is the best course of action if they don't do anything about issues involving other coworkers. If they don't care enough to do anything when it is reported, they are not going to do anything if you give an ultimatum other than making your job miserable enough that you quit or fire you, whichever costs them less.
Don't under perform in a way that gets you fired though, as that could jeopardize any money you get from being fired like unemployment or whatever you have in your location.

Whatever you plan on doing if it involves a company, make a paper trail. You don't want their defence to be a reliable "Nuh-uh!! Did not!!"

You should always point out abusive behaviour but making ultimatums is not a good strategy, unless you're ready to take the nuclear option regardless.

If HR and manager do nothing to address the problem, wouldn't it be a better strategy to start working the least possible and let the company fire me, while looking for another job?
Solid tactic.

It's only a good idea if you believe they are waiting for a final reason to fire the person.
I'd go for another job if it's not the case. Sometimes a workplace just isn't for you.

Depends, do you like the job ? Is it worth the effort in changing the dynamic?

I think doing a good analysis of strategy here will depend on a lot of factors.
Firstly, before coming up with a strategy, it is good to have a clear idea of your goals / the strategic problem you are trying to solve. I see or could infer a few possible ones: you want to work in an environment where you don't feel bullied, you want to ensure others aren't bullied, you want to see bullies punished, to maintain positives in the company and want to enjoy those without the negatives of being bullied, or perhaps that you believe in the goals of the company or have some stake in it, and want it to succeed. Different goals might lead you to a different course of action.
Next, you would want to diagnose what's really going on. Are there just a few bullies, in a company mostly full of professional people, or are the bullies the majority? Are senior leaders in on the bullying, or is it only lower level employees? Why do you think the bullies were hired in the first place - is it because bullying is considered okay in the company, or is it not considered okay but they slipped through? Why do you think the bullying hasn't been addressed already? Is it because senior managers don't know? Are the bullies friends / relatives of senior leadership? Are the bullies high performers that the company really would want to keep around, or do they get barely get anything done? Also, are the bullies even aware they are being bullies? Are they unaware they are being insensitive, and likely to change if made aware, or are they actively being malicious and well aware of the impact?
Next, consider the direction you want to take, and analyse the likely impact on your goals. You could find another job - how easy that is would depend what the job market looks like for your role, and how good the terms of your current job are. It wouldn't achieve goals around making it better for others. You could try talking to the bullies if you think that they might just be unaware of the impact of their behaviour and that they might change. If that doesn't work, you could try talking to a manager / HR member, perhaps either to arrange mediation, or for them to take action. You could also just try ignoring the bullying if it isn't having much impact.
To choose from the many possible directions, it might help to think from the perspective of the company shareholders, senior leadership, and HR department. What would you do in their shoes if you learned of the bullying? If it is the majority of the company doing the bullying, then something like replacing all the bullying staff is going to be an instant non-starter. The best possible would be to slowly roll out training, policies, and new hiring practices to try to improve the culture over time. If it is a few people who, it now turns out, are the reason for high staff turnover and lower profits, then they might be quite happy to take action. Although probably not if the bullies are the senior leaders.

Examples? And how often is it happening?

Idea is good, in principal. Can I just offer some thoughts to get things happening smoothly.
Bullying, what type? I can only offer general overall advice without specifics. So I'll offer some food for thought, What if that person is neurodivergent and doesn't understand they're bullying, or coming across too blunt. I would suggest, address the behaviour directly, succinctly and without judgement.
Talk as if they're an alien and don't understand our culture, and you have belief once they understand they will step up. But say it as plainly, in the least amount of words possible. Don't leave room to debate, just straight plain facts. This causes this. Please don't.
Ask how you can help them be more effective in positive engagement.
On a personal note it's always a good idea to listen to the messages your body is giving you, those emotions are a need your body has, it isn't about anyone else, it's about your lived experience and how comfortable you are in your environment. Times you haven't felt safe can be a trigger for you in the future, and then bullies words hurt more. If you feel you have unhealed trauma or a low sense of self and self worth because the world in general has chewed you up, (which it does so often it's almost a universal experience) find and build on yourself in those areas. It doesn't minimise that the bullies are harmful, just utilise their efforts to find places you can up skill and heal, if they are being highlighted for you. They still suck. You just take what they give you and turn it into a tool to build on yourself and build yourself up. Be genuine, curious and open and deep dive on your feelings. Eventually, after studying what's freely available out there and uplevelling your sense of self, you start to see all of the bullies behaviours are entirely about their own broken insides and damage. It's really hard to be upset by words from someone you pity.
Avoiding toxic environments is a better way to live. Finding ways to make that environment non toxic is a good life philosophy. Do your labour laws already have protections built in to protect you from psychological abusive type behaviours in the workplace?

you should already be doing the bare minimum. no job pays enough to buy back what it takes from you.
but as has already been said have something lined up before you go to management with this and don't even hint that you're thinking of walking.

HR and management will never help you.
You should bully them back.
Offer to bring everyone coffee from the place they all like. Add a little powdered laxative. Do the same thing again next week but with more laxative. Gradually up the dosage until everyone has the mega doodoo blastoff and accuses the coffee shop of doing food poisoning on them.
Put low batteries in the smoke detector so it constantly beeps. When someone changes them, put another low set back in the next day.
Apply the “when vestmoria is offscreen, other characters should be talking about vestmoria” principle to meetings and presentations.
Always keep a web browser window open to some gun manufacturers website. When someone walks up, alt-tab it to the top then alt tab it away at the last minute.
If you’re not doing it already, butt into conversations and start mansplaining to everyone about the thing they’re talking about. Bonus points if you’re a woman.
- Don't do the laxatives things. Poisoning people is very illegal, for good reason.
  
  Bring a bigass eight line hookah into work and blow banana flavored clouds. When people complain offer to switch to a different flavor.
  
  If you maintain documentation or comment code, start weaving a disconnected, corkboard narrative into it.
- This is the worst advice possible haha you'll end up getting arrested.
  
  Buy an old carbureted v8 sports car or pickup truck. Get to work early every day so you can leave early as well. When it’s time to go, start revving the engine “so it’ll warm up and even out” in the parking lot right next to the office.
- What is actually wrong with you?
  
  When people talk about the news in gaza around the watercooler, always defend israels actions and bait people into accidentally saying things that you can accuse them of being an antisemite for.
- I was with you until the gun thing
- Every single one of these things is something that will obviously point back to you.
  Some of these things not only could get you fired but hauled away by the police.
  So... no.
  
  Eat pea soup for breakfast and rip huge farts in the office.
  Punctuate each one with “that was me” or “that was you”