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 $X$.

中文翻译：

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半代数集的实热带化与分析

令 $K$ 是一个具有非平凡非阿基米德绝对值的实闭域。我们研究了热带化地图的改进版本，我们称之为真正的热带化地图，它考虑了 $K$ 上的符号。我们从一般的角度研究这张地图下 $K^n$ 的半代数子集的图像。对于半代数集 $S \subset K^n$ 我们定义一个空间 $S_r^{\text{an}}$ 称为实分析，我们证明它同胚于 $S$ 的所有实热带化的逆极限. 我们证明了热带基本定理的真实模拟，并表明任何半代数集的热带化都可以通过在半代数集上有效的有限多个不等式的热带化来描述。我们还研究了真实分析和热带化的拓扑特性。如果 $X$ 是一个代数变体，