I don't think it's any harder to wrap your head around than the notion of parallel realities. Consider:
1) We routinely accept alternate versions of characters can exist in other realities.
2) The notion that making one decision or another can at any time create
a parallel reality is not new.
So splitting one person into two, with neither having the better claim on "realness", is pretty standard stuff. The only real difference here is that the two versions can interact routinely.