The intersection of zero loci of a family real analytic functions

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 “analytic” 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
$$K cap bigcap_{i in I} V_i = K cap bigcap_{i in J} V_i ,?$$