If $(f_i)_{i \in I}$ is a family of holomorphic functions on some open set $U \subset \mathbb C^n$ with zero loci $V_i$, then the intersection $\bigcap_{i \in I} V_i$ is complex analytic. Moreover, for every compact $K \subset U$ there exists a finite subset $J \subset I$ such that $$K \cap \bigcap_{i \in I} V_i = K \cap \bigcap_{i \in J} V_i \,.$$ See Is intersection of zero set of any family of holomorphic functions an analytic set?
Now take the same setup but replace "holomorphic" by "real analytic" and $\mathbb C$ by $\mathbb R$.
The first statement is then trivially true, or at least if $I$ is countable: define $f = \sum_{i \in I} c_i f_i^2$ for very rapidly decreasing $c_i>0$, then the zero locus of $f$ (where it converges) is $\bigcap_{i \in I} V_i$.
Is the second statement also true? That is:
Does there exist for every compact $K \subset U$ a finite subset $J \subset I$ such that $$K \cap \bigcap_{i \in I} V_i = K \cap \bigcap_{i \in J} V_i \,?$$
