Countable field which is elementary equivalent to $\mathbb{R}$

Question

Is it possible to describe concretely a countable field which is elementary equivalent to the field $\mathbb{R}$ of real numbers? Such a field exists by Löwenheim-Skolem.

Answer 1

The field $\overline{\mathbb{Q}}\cap\mathbb{R}$ of real algebraic numbers is elementarily equivalent to $\mathbb{R}$. This follows from the fact that the theory of real-closed fields is complete, and $\mathbb{R}$ and $\overline{\mathbb{Q}}\cap\mathbb{R}$ are both real-closed fields.