Suppose that I'm given a set of rational primes $S$ with positive Dirichlet density, and a finite set of primes $R$, disjoint from $S$. Does there exist a number field $K$ that is;

- unramified outside $R$;
- splits only at primes in some subset $S'\subseteq S$?

In my situation, I have a semisimple representation $\rho$ of $G_\mathbb Q =\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)$, unramified outside $R$, and I have information about the action of Frobenius elements $\mathrm{Frob}_p$ for $p\in S$ (let's say I know their characteristic polynomials). I'd like to know if this completely determines my representation on some open subgroup of $G_\mathbb Q$.