"But for them to compare notes, Bob has to return to Earth. "
Not really. If this entire thought experiment relies on Bob turning around to compare notes, how about if they had an instantaneous quantum communications device and they could compare notes without Bob having to turn around? Then how do you solve the problem of who is moving? Seems the solution is, during communication they would be able to tell who is going faster and who is going slower just from the rate of communication in each direction. The communication itself may be instant, but the flow of data should be different depending on who is sending.
Yes, but we don't even need this quantum comm. device. We know the distance to the travelled star, thus Bob could radio the time he reached the star and Alice would simply wait the number of years necessary to read it and subtract those years from Bob's time value. IOW, it would be clear that Bob arrived much sooner due to relativistic effectss.
The asymmetric approach to explain the paradox is quite unclear. SR can be used in accelerating frames by taking each frame individually and adjusting. Also, if the universe were to suddenly travel in the opposite direction of Bob, then he would experience the same thing as that of his actual deceleration efforts. So, why would Alice be stuck holding the long-time stick?
Another stickler is that Bob may have arrived to a distant star in a time that is years faster due to SR. But if it only takes him, say, a week to slow down, then we are to somehow think this will cause Alice to "lose" all those years in 1 week?
I wish I understood this.