Thanks for all the comments, and sorry for my extended absence from this thread. To cut a long story short, the culprit was that Fresnel material. It has its faults, but I've got away with it in long shots before - and not so long, my avatar's glasses are made the same way. He's based on Michael 2. I'm sure I've also used the same material on generation 3 figures' glasses without trouble. There's something about V4's eyes that seems to bring the problem about, but I haven't tracked down what it is yet.

I used a material which I've called "BB's 5 node Fresnel" - I think I got it from the RDNA forum - and that works fine.

I'll post back when I know more. I have a rather ghoulish test set-up consisting of a single V4 eyeball sitting in front of a sheet of glass. :)