'Ask HN: How do you fix engineering culture?'

Based on your question, I would start with a mirror.

'but can't keep engineers'

So why are they leaving? Although it could be pay, it is doubtful the prime issue.

'An engineer left us three weeks ago and two more gave notice and are leaving in January.'

Happy people do not look for jobs in December, well not in the states. My guess is, these people must have been very unhappy, and the joy of the season cemented their dislike for your company. Have you looked on glassdoor to see what the employees are saying? I have considered working at two companies that were considered wonderful places to work at but the techie section was considered a hell hole. One of them I really wanted to go to because it just seemed like the perfect company. The other glassdoor told me all I needed to know. Needless to say I did not bother. I suspect it was the engineering management that made life suck there.

'We have a strong culture, but it is widely felt by the rest of the company that the engineering team doesn't adhere to it.'

I read this as, we have a 300 page employee manual and the techies ignore it.

'decision they made to isolate themselves in their own building'

Sounds like a trust issue. Or is it a noise issue? Personally I would never take a job in a noisy environment, and most good techies I know of will not either. Of course they ALL love their own noise... thus the headphones, no speakers rule.

'actively engage in bickering and in-fighting between their own teams'

bickering can be helpful, but the infighting implies a lack of trust in the reward system you have provided.

'extremely limited communication to the rest of the company (users of their products)'

Clearly management failure. But not necessarily Engineering management failure.

'reports of sexual harassment and racism'

in the current environment this can cut either way. You need to look at your turnover. People being harassed for sex, race, other are very motivated to leave. People falsely accused are as likely to leave also. Which ever the case is, you must root this one out to avoid more problems and possible lawsuit.

'There's zero consistency between the message they're sending'

I think your company is not exploring the complaints well enough.

'Some engineers believe the team is moving too fast, while a different group believes they're moving too slow'
Yes, I can guarantee that is true. If your management sees this as a problem, then you do not have good management. Or maybe they are not technically competent.

'complaints of micromanaging mixed with complaints of lack of management'

Typically I will complain about this when my management is not technically competent but thinks it is. Generally this is the management is micromanaging me, but not managing those other idiots.

'new flagship application we're building is extremely simple'

You said you are not a manager of the engineering team. But are you an engineer? How do you know how simple it is. Since you know how EXTREMELY SIMPLE it is, why don't you wipe it up this weekend in your spare time? This is a pet peeve of mine. I have had more than a few coworkers who felt everybody else had easy tasks. Amazing when offer to trade work with them they shut up.

'development process is indeed slow and full of fail-safes, but there's still an astounding number of bug reports for a simple app'

Depending on the application it could be reasonable. On the one hand you can have testing generating a lot of bug reports, because the OPINION is different. I have worked at places where we would get bug reports where 30% of them were 'above keyboard errors'.

In the end I see 3 possible fixes:

1) new management - both in engineering and in other departments. In engineering you will need a more technically competent leader. Many engineers will stay to keep learning from a good leader. Other departments need to be lead by people who understand working with engineers.

2) re-examine your reward system. I suspect most feel it reward the 'suck ups' and not the real workers. Because it sounds like your real workers left.

3) try the golden hand cuffs. Many people have stayed for that. But they do not have a passion for the work.