Turns out, yoga mats make the best play surfaces ever! they hold the models in place perfectly. Novel concept considering these were designed to have a large amount of friction. They place just enough grip so they don't slide around...
This sounds great, and not too expensive--I'm finding oversized yoga mats at 36"x72" going for $15-$20 all over the place online, which means you could split the cost with a friend and get a high-quality mat for under $10.
...and keep dice from going all over the place.
I know it's slightly OT, but if keeping your dice corraled is a concern, you want a dice tray or tower. Rolling dice on a non-slip surface actually decreases the randomness of the result!