As a simplified, purely mathematical answer, the signature of the metric is baked into the initial/boundary conditions. First of all note that it cannot change along a smooth enough connected space-time, as regions of different signatures would need to be separated by sets on which the metric is degenerate, i.e. singularities where GR does not hold. So once the signature is fixed at some point, it is fixed everywhere.
Now consider the kinds of problems we would like to solve:
If you want to use GR to predict future behavior of something, e.g. of our solar system, you would start with its current state as initial data and continue from there. We know that the current state has a locally Minkowski metric. But then all future states have the same signature.
Another type of problem that is studied abstractly a lot, is the influence of gravity on the space-time around a heavy object, e.g. the Schwarzschild-solution. We think of this object as existing somewhere in our universe. Even though we do not model the rest of the universe, we would like the solution in empty space far away from the object (i.e. the boundary condition at infinity) to behave similarly to the empty space we know. So the boundary condition again specifies a Minkowski signature.
The same applies to all other problems. If the signature of your solution is different from the signature we have at home, there is no way to find a space-time that connects your solution to us. Whatever universe you are calculating, if the signature is wrong, it might have some mathematical interest, but (at least within GR) it is completely inaccessible to our physical reality and thus strictly speaking not physics.