Almost all reals are not computable, but as far as I know min is total - given two reals (represented as e.g. lazy infinite decimal sequences), computing their min is trivial.

Yes, I'm wrong. I was thinking of the naïve implementation

     if ( a < b ) then { return b } else { return a }

which requires being able to decide order, but, as child comments point out, that's not necessary.

Unless when they happen to be equal. Then it never halts.

Actually, computing the minimum should work: If they are equal you can still lazily produce the digits of the decimal expansion. What you can’t do in this case is to tell whether this minimum is equal to the first or the second number.