Thanks for the fast responses, guys!
To answer some questions:
I am spraying water-based Auto Air Color paint.
I'm almost positive it's not orange peel, although the photos might not clearly show that. It really just looks like the paint won't stick to certain areas as it goes on. In fact, some of it looks almost like a wall in a house that gets mold or mildrew on it.
I have a good HVLP gun and I've changed the PSI at the gun (it's 40 at the compressor), plus I've fiddled with all the other settings, but nothing helps. Before this started to happen I was getting nice smooth coats. This is the fourth coat to give me this problem now and I've been going crazy trying to figure it out.
The silicon diagnosis sounds the most likely. I had a bucket of clean rags I was using to wipe the body with before I used the tack rag, but I have a strong feeling that my father-in-law used a few to clean his car and then dumped them back in the bucket. I actually could smell some kinda armour-all-type smell on them, but I put it down to something else at the time.
Hmm. I understand about doing the dust coats, but should I still degrease the body first? If so, what can I use to degrease the body with that won't eat the water-based paint?