yeah pretty much everyone is having this issue from what I gather. Someone on the steam discussion posted a utility that lets you up the RAM cap on 32 bit executables (which BE is), from 2 to 4. Apparently the reason it happens is basically BE runs up the consumption to over 2 then you start getting all the black squares mess and start losing the effects
Anyway I made use of the utility and it didn't fix the problem, but it did delay it. I went to 138 turns (and reached Harmony 8) as the NSA on a standard frigid map before stuff started to majorly bug out (7 civs instead of 8, which is what I usually play with). I made use of T calculus and went heavy on exploring and fighting to test out the animations over and over while filling in a ton of map.
did you know that level 3 gunboats shoot missiles if you use a ranged attack right next to the target? I didn't, now I do. that was also one of the first few things that bugged out.
Anyway, I recognize just one attempt with it could be a fluke. The original guy says he tried to use the thing on it but it crashed his game on startup. so either I did it wrong (and it really is a fluke) or he did it wrong. Tool's here if you wanna give it a go:
http://www.ntcore.com/4gb_patch.php run it, select BE's executable, patch it, run game, ????, temporary profit.