OK, last night I spent a little more time trying to track this problem down, and I did. I also suspect that this problem manifested itself in an "ugliness" that I hear when I use Amplitube software in some situations. It's the EMG-85 humbucker that's in the bridge pickup! Here's the guitar:
http://rockbeareguitars.com/electrics/steinberger/89-gm4t/
Mine's white, but that's the deal. The sound completely went away if I use the middle pickup or the neck/middle. I did have the bridge pickup pretty high, and backing it down did make the problem almost totally go away. I don't know if this could be part of the issue, but I have a Sustainiac Stealth Pro installed, so that's replaced my neck pickup. BTW, that pickup as a really cool sound when used as a pickup, but that's another story.
So anyway, my adjustment brought the problem down to nearly zero, and a quick pickup change eliminated it totally. I guess I can live with this as it only seems to happen on some clean models anyway, but this schmutz was probably there the whole time, though masked by distortion.