I encountered a similar question while studying model theory. One can prove that, if a sentence in field theory is satisfiable modulo infinitely many primes $p$ (i.e. in conjunction with the sentence $1+1+\dots+1=0$ with $p$ $1$'s), then it is satisfiable in some field of characteristic 0.
By taking the sentence to be "this set of polynomial equations with integral coefficients has no solutions", and taking the contrapositive, one proves that, if the set of equations has solutions in all fields of characteristic 0, then it has solutions modulo infinitely many primes.
But this is not very useful, since if the equations has solutions in $\mathbb Q$ then it already has solutions modulo infinitely many primes. So, the question naturally arises: Does having a solution in $\mathbb C$ or $\mathbb R$ imply the existence of $p$ (and further, the existence of infinitely many $p$) so that the equations have a solution modulo $p$?