Document. Clearly state your goals for them, and have them written down in a 1:1 document.
Communication style and tone are important, and autistic people do not get a pass on trying to understand them, even if you may want to give them more grace when they slip up.
With that in mind, a goal should be something like “coworkers report improvement in communication style and tone.” It should be something that other people assess, not a checklist of things for the employee to self-report progress towards. (You never want to give people who don’t correctly see how they’re perceived by others the option of grading themselves. First, their lack of self-awareness means that they aren’t the best judge of improvement, and second, treating things a a checklist to complete will give the impression that you wanted some line items addressed, rather than a holistic improvement in a vague problem area.)
When you document, pay special attention to times when their misunderstanding caused rework. Use those as examples to help them improve. It can be tough for autistic people to take a vague problem and figure out a plan to tackle it, so the more specific examples of problem, outcome, and desired change you can make for them, the clearer it will be what is expected of them. It is better for you and for them - they have a better chance to improve, and you have documented instances of business impact if a turnaround doesn’t happen and you need HR to sign off on moving on.
For accommodation - they have not told you that they have a disability, you’ve heard it secondhand. It’s their choice to make you aware of their autism, and also their responsibility to request accommodations since they have not told you yet. However, you can make informal accommodations, but I would treat it the same way you accommodate employees’ individual personalities and strengths/weaknesses. They, like everyone else, should not be below a certain standard of behavior, but within that standard there will be differences in what each individual needs. You employee sounds like they need direct coaching and clear examples of what is or is not working. Try to give them tasks with high clarity and limited communication requirements, if you can. However, if you find they need more clarity than a reasonable person would expect, again I would have a goal of “the PM reports a decrease in the amount of back-and-forth on tickets you pick up, without a drop in velocity” or something similar. If they find they need more formal accommodations than that, well, now is the time for them to tell you about their autism diagnosis.
Best of luck! This is a challenging managerial task.