@WillSawin Yes you are absolutely right. This seems to make the anaytic continuation part rather simple in descriptive set theoretic sense...

Are Tarski's and Hilbert's axioms that different? I thought they are bi-interpretable and have the same models (excluding continuity axiom from both, of course)

What exactly do you mean by the spirit of Tarski? There are several books that develop synthetic geometry based on first order logic (or at least it's straightforward to phrase their exposition in FOL), such as Hartshorne's Geometry: Euclid and Beyond, John Lee's Axiomatic Geometry, and Francis Borceux's An Axiomatic Approach to Geometry.

There is even a continuous $f$, using a continuous open map $h:\mathbb{R}^3\rightarrow\mathbb{R}^4$

@NoahSchweber Could you indicate how to show (or why we should expect) that this equivalence relation is coanalytic?