Help me track down a noise issue

[GuitarStv]

Does the tape from each of the pickup cavities need to connect to a common ground and to each other? I guess a necessary first step is taping the cavities, but I'd rather not have to restring multiple times.

Yes. All shielding in a guitar needs to connect back to your output jack ground. That's actually how I check if I've shielded stuff correctly - put a probe on the shield and on the output hack ground (which is the sleeve - tip is hot) and you should get 0 or damned close to it.

If you don't do this, the shielding that you put in your guitar will actually act as an antenna and can introduce noise.