Both X and Y have their zero and one elements.
Add the one element to itself over and over to form the naturals in X and the naturals in Y.
Include the additive inverse of each natural to get the integers for both sets.
Use the multiplicative inverse of each integer and closure under multiplication to get all the rationals for X and Y.
Use the Cauchy sequence construction within X and Y to get limit points for all the other reals.
Or maybe once you have a correspondence for the rationals in X and Y, that is enough to pin point the reals between them.