For example, maybe in a FPS, part of the non-gameplay-critical graphics use particle generators for a cool effect that not all players see (because it's behind a building for some of them and thus doesn't even need to be rendered); if these generators used a synchronized RNG, then all players would have to do computations for every particle effect happening anywhere, just so that the combat and more game-important RNG values would be in synch when they really need to be.
If you're using a global random pool, you've descyned here unless all players have the same thing on screen.
Once it comes time to simulate that next turn, if you have something different than other clients because of a missed update or graphics lag, even if object positions and the random seed going to be "fixed" by another turn update, all future interactions with any objects that over- or under-sampled random will be wrong and could create further sync problems.
Can anyone grok this? I can't see why this (each player's simulation making the same number of calls to random) would ever not be the case if all players are running the same patch of the game and are executing the same commands.