I've done some studies with the Fade, though I'm still far from an expert. I want to be clear that I'm moving into theory, here, a bit. But I know I'm not the only one who thinks the most likely thing is that the Fade touches people's minds. Not only in Thedas, but in other worlds. Independent of chronology, clearly. And your original self is still where you were.

People in Thedas have reported encountering people who have died, in the Fade. Some cultures here think the Fade is a place souls move through on their way to some other afterlife. Other think they stay there. Evidence suggests to me that, at least as a rifter, my memories of my time in Thedas before lingered in the Fade somehow. So this second time, it drew not only my memories from home, but also from my first time in Thedas. That said, I suspect my body ... I don't think it's continuous. Haven't found a good way to test it, but I vanished last time. And I came back well, even though in Thedas I hadn't been cured of the disease I cured at home.