How to politely inform potential PhD supervisor that I must wait before responding to an offer? (Two-body problem)
In my opinion, since you are already accepted, there is no harm in telling about the two-body problem; if you end up going to University B, Prof@A.edu would likely feel better about that reason than some negative impression of them personally or of University A.

You are well within your rights to take the full 30 days to decide. Certainly not all professors are equally versed or sympathetic towards the two-body problem, but most have experienced it, if not personally, through their colleagues, friends, etc.

Ultimately I think it is your choice whether or not to explain your reasoning, but I would personally lose a bit of respect towards your Prof@A.edu if they were unable to accept you stating simply "My offer gives me 30 days to respond, and I would like to take that entire time to ensure I make the right decision for me. I will inform you immediately of my decision once it is made."