Hmm...that makes me want to give Brian Greene a kick in the pants.
What he conspicuously *fails* to mention in the above is that the alien *cannot exchange information* with *any* of the events on the Earth's worldline that could be in his *now slice* by him changing his state of motion. He is spacelike separated from *all* of those events, so there is no causal contact between them. So the part in bold above is simply a non sequitur; the only sense of "past" and "future" that would justify saying that "the past must be real, the future must be real" is causal past and future, and the alien and Earth are *not* in each other's causal past or future, no matter how they move.
Put another way, take any event on Earth's worldline that the alien could label as part of his "now slice", and call it event E. If event E is in the past for you, on Earth, then you, at your "now" instant, know what happened at that event, but the alien *can't* possibly know. And if event E is in the future for you, on Earth, then you can have a causal influence on that event but the alien can't. So either way, it doesn't matter which event the alien labels "now", or whether it's in your future or your past; whether or not that event is "real" has *nothing* to do with the alien at all.
I'm disappointed that Greene muddies these waters, but unfortunately I'm not surprised. I remember having a similar feeling a number of times while reading one of his books (Elegant Universe or Fabric of the Cosmos, can't remember which). I've seen much better discussions of how spacetime works in other popular science books; IIRC Penrose has a good one in one of his (again, can't remember which), along the lines of: your "past" is your past light cone, and has already happened; your "future" is your future light cone, and has not happened yet; the rest of spacetime is "elsewhere", and you can't exchange information with it either way so there's not much point in worrying about it, or trying to pin down where exactly "now" is within "elsewhere". It's just not a big deal.