Which quantum reality will be seen in a future Earth
All of them.

If you sit on Earth then (in the many worlds interpretation) each quantum mechanical observation creates multiple universes, and there is a copy of "you" in each one. So if I were to ask, which universe will I be in tomorrow, the answer is "all of them" (except the ones in which I die)

If I go into space, float around and come back I will come back in all of the universes. Because I have been moving (relative to Earth) there will be a slight clock discrepancy in all the universes.

If I go into space, orbit a black hole, then come back. I will come back in all the universes (except for the ones that I die in). There will be a large discrepancy in clock time but that is irrelevant. I've not done any movie time-travel. It's just that my watch doesn't match the clocks on Earth. 

In the Copenhagen interpretation you come back in the one universe (with a certain probability that is close to 1). As the mathematics is identical, there is no experiment that can distinguish between different interpretations of QM.