What does a Real Number Object look like as a sheaf?

Question

I understand that a Natural Number Object, when considered as an object in a sheaf topos, consists of a sheaf whose sections are locally constant functions. This makes me wonder what a Real Number Object looks like. I'm aware that there's a distinction to be made between Cauchy Real Numbers and Dedekind Real Numbers. However I'm not aware of what either type of sheaf has as its sections.