Without any more information than "trouble" or knowing the guy, I can't give you any specific advice, so I'll just share what my experience is like.
The longest I've ever been able to hold a job was about three years. Arbitrary rules and meddling management is like torture to me. Corporate culture is utterly intolerable. I tried for a very long time to fit in, only to burn out dramatically every time. It's a cliche, but being confined to a tiny, fixed role as a replaceable part in a corporate machine feels to me like caging a bird. I can do so much more and it brings me great suffering to be confined that way. I need to be able to use my talents and do what I'm good at, it's the very core of what I am, it's my primary driving force. To the point where it becomes an obsession, a compulsion that I can only barely contain.
A large majority of autistic people have ADHD, which has the unfortunate effect of making them much more intense. I forget meetings constantly. Deadlines are just a passing thought because I have no clue what day it even is. I start more projects than I finish, and I rarely work on a single project for more than a few days.
But when I get my shit together and actually focus on something? Holy shit, you would not believe the amount of work I can crank out. My present job is as the R&D department. Just me. We should have three people doing my job, but I am an absolute force of nature once something catches my attention.
I have trouble talking to people. I don't make friends easily, though almost everyone likes me. Unless someone specifically asks, I tend to not offer details about my personal life, and I generally don't ask about other's. That's only because it genuinely never occurs to me to ask until the conversation has been over for days. I can't do all the social niceties. I don't have whatever invisible rulebook normal people have for social interactions. I do my best to imitate, but it's honestly as complex as anything I do at work. Most days I simply don't have the brain space for it.
When pushed, I can be an eloquent presenter, a respectable and personable engineer, I can talk to hundreds of strangers at CES and show them the demo. I can do or be whatever is needed, it's just a different mask. But the cost is very steep. After three or four days of CES, it took me months to recover. After I presented a panel to a packed crowd, I nearly passed out. When I have to write a Serious document, I get anxiety attacks for days. It's led to a real primal fear of these kinds of activities, and it's becoming a problem that gets worse as I get older.
Sometimes I say inappropriate things when I'm stressed. The composure I project is just another mask I wear. Underneath I can be cruel and vindictive, spiteful and petty. It comes from a lifetime of trauma and it's taken me a very long time to deal with. When I'm stressed, my control slips and I say things I don't mean and become lifelong regrets.
Fortunately, I've found a good job with a tech startup. My boss is very good to me, which is why I'm able to shoulder so much work. He doesn't micromanage, but will nudge me if I miss a meeting or a deadline. He pushes me to improve but will accept no if that's outside my boundaries. When I say stupid shit he brushes it off. Generally, he understands that I'm going to do what I'm interested in, and do a lot of it. I can't be controlled, but guided. All he can or needs to do is point me in a direction and let it happen.
Fundamentally, understand that autistic people are just people. We want to fit in and get along as much as anyone. And if you're on a part of the spectrum where you can be a successful developer, you're probably very aware of your differences and limitations. But no one gave us the book that the rest of you play from. We need clear, well-defined rules with logical, coherent reasoning behind them. If rules don't make sense, they'll be broken when convenient. But if you punish me for breaking a rule I didn't know about, you might as well have killed my dog. Absolute mortal enemy, never forgive, never forget.
Being autistic in the professional world is more difficult than you can imagine. All of us have deep, serious trauma from growing up in a world you can't understand and that won't understand you.
Talk to your employee. Be respectful and try to understand. Don't play word games and don't leave things unsaid and implied. If you don't say something, an autistic person is not going to hear it. Lay out the rules, explain your reasoning or offer an honest "I don't know". If behavior is inappropriate, clearly say so and define what would be appropriate. You can't just say something is wrong with no explanation, it comes across as random unjustified punishment rather than a correction. Things must be logically consistent and have reasons that make sense. Otherwise, you're going to get resentment and burnout and nobody is going to be happy with the results.
Correcting behavior with an honest conversation will work and earn you respect. Punishment will never, ever work and is more likely to blow up in your face. He doesn't want to be a pain in your ass, he wants to get on with his work. What he wants is most likely to come to work, solve interesting problems, have a nice day and go home. That's really it.
You'll have to just accept his limitations. If he's a poor communicator, don't make him speak up in groups. If he can't turn in paperwork, you might have to work out a system suited to him. Unfortunately, it will probably be all up to you to identify problems and start the conversation about solutions. If you asked him what he struggles with or needs help with, you probably won't get a good answer. It's like being color blind, you never know you have a limitation until it's pointed out. You will have to make some accommodations, but that's just what life is like with a disability. However annoying you find it, imagine what it feels like to see everyone do something that you just can't do, no matter how hard you try.
Two final points: after you talk with him, he will likely be very stressed out or upset. Don't read into it. It's a uniquely stressful situation to have to be corrected that way, but he'll take it to heart. If he's anything like me, he won't be in shape to be around other people for a while. Offer him a couple days off, or work from home. You may need to gently insist, but do take no if he means it.
Also, never tell him about this situation with HR tiptoeing around him. If he doesn't know, it's far better to leave it that way. Finding out about something like that would destroy me. I would honestly quit my job and never return. To any person with a disability, that kind of thing is either mortifying or enraging. Either way, absolutely no good will come of it.