Real tropicalization and analytification of semialgebraic sets

Let $K$ be a real closed field with a nontrivial non-archimedean absolute value. We study a refined version of the tropicalization map, which we call real tropicalization map, that takes into account the signs on $K$. We study images of semialgebraic subsets of $K^n$ under this map from a general point of view. For a semialgebraic set $S \subset K^n$ we define a space $S^{an}_r$ called the real analytification, which we show to be homeomorphic to the inverse limit of all real tropicalizations of $S$ and to satisfy a real analogue of the tropical fundamental theorem. We also study its topological properties. If $S$ is an algebraic variety, we show that $S^{an}_r$ can be canonically embedded into the real spectrum $S_r$ of $S$, and we study its relation with the Berkovich analytification of $S$.