Yeah, that was the only idea I had, maybe the new Catalyst drivers are causing it. It's a pain in the butt, but I guess I'll try rolling back to the older ones I was using. I have to uninstall the new ones, reboot, install the older drivers (which fortunately I have tucked away on my hard drive) and reboot again.
UPDATE: Turns out it was easier than that - I just used the "Rollback" feature in Device Manager. Now I'm using the 6.4 Catalyst drivers again, and guess what? The asteroid texture problem disappeared!
Some "upgrade" that turned out to be! So much for religiously updating one's drivers.
There was another problem the new Catalyst drivers caused, which I didn't think to mention before as it was relatively minor: any time I opened a dialog, such as when I clicked Exit on the main menu or pressed [ESC] in-mission, everything except the dialog took on a weird blue tint. Doesn't happen now that I'm using the older Catalyst drivers again.