Is There a Global Time Flow?

Causality conditions eventually allow you to say that we can basically put coordinates on all of spacetime. We don't have just little patches for different patches of spacetime, is that correct? Well, it's not really coordinating, because coordinates is something that's related to the very concept of a manifold. But usually there is no single coordinate chart which can cover your entire space. Like, given on a sphere, you cannot really cover the balls at the same time. And that's the same story about the spacetime or a physical universe, if you wish.