  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
$S_r^{\text{an}}$ called the real analytification, which we show to be
homeomorphic to the inverse limit of all real tropicalizations of $S$. We prove
a real analogue of the tropical fundamental theorem and show that the
tropicalization of any semialgebraic set is described by tropicalization of
finitely many inequalities which are valid on the semialgebraic set. We also
study the topological properties of real analytification and tropicalization.
If $X$ is an algebraic variety, we show that $X_r^{\text{an}}$ can be
canonically embedded into the real spectrum $X_r$ of $X$, and we study its
relation with the Berkovich analytification of $X$.
