Nothing in the construction requires you to show an algorithm that given x, y \in R allows you to compute x+y and xy. You would make the usual Dedekind construction and show it satisfies the axioms of such a field. (as a matter of fact, no such algorithm exists in full generality!)

It's probably a tomato/tomato kind of thing, but I'm only objecting to the 'algorithm' part of parent's comment.