One easy definition of the reals is that each 'convergent' sequence of rational numbers defines a real number. Two sequences define the same real number when the sequence of differences converges to 0.

In this setting, the problem of deciding if two real numbers are equal is the same as deciding if a specific sequence converges to 0. Apparently there are 'reasonable' sequences for which this is an undecidable problem.

I'm afraid I can't give you two reals whose equivalence is undecided. I expect it will involve some rather weird descriptions of real numbers. I would love if someone could give some examples.