The most abstract version of the Tarski-Seidenberg theorem I know of is the following
Let $f:A\to B$ be a morphism of finite presentation of commutative rings. Then the induced map
$$f^*:\operatorname{Sper}B\to \operatorname{Sper}A$$
sends constructible sets to constructible sets.
Here $\operatorname{Sper}$ is the real spectrum of the ring, i.e. the set of all pairs $(p,<)$ where $p$ is a prime ideal and $<$ is an order on the residue field at $p$.
It is well known (e.g. theorem 7.2.3 in Bochnack-Coste-Roy Real Algebraic Geometry) that if $A$ is an algebra of finite presentation over $\mathbb{R}$, the boolean algebra of constructible subsets of $\operatorname{Sper}A$ is in natural bijection with the semialgebraic subsets of the real points of the variety $\operatorname{Spec}A$.