There are many different types which are called real numbers in the literature. These include:
Dedekind real closed intervals, where the Dedekind real numbers are the located Dedekind real closed intervals, and the computable real numbers are the Dedekind real closed intervals with a locator.
Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)
Mark Bridger, Real Analysis: A Constructive Approach Through Interval Arithmetic, Pure and Applied Undergraduate Texts 38, American Mathematical Society, 2019.