Well, models can have the same reals by fiat. If I cut off an existing model below an inaccessible, I certainly haven't changed the reals. Alternately I could restrict to the constructible closure of the reals L(R), which satisfies ZF but generally fails Choice (you don't expect to have a well-ordering of the reals in this model).

I think, though, that Stuart_Armstrong's statement

Often, different models of set theory will have the same model of the reals inside them

is mistaken, or at least misguided. Models of set theory and their corresponding sets of reals are extremely pliable, especially by the method of forcing (Cohen proved CH can consistently fail by just cramming tons of reals into an existing model without changing the ordinal values of that model's Alephs), and I think it's naive to hope for anything like One True Real Line.