A classical closure of a field of characteristic 0 is either an algebraic closure, a real closure, or ap-adic closure for some rational prime p. We say a field is pseudo classically closed if it satisfies a universal local-global principle for rational points with respect to finitely many classical closures. Following the extensive body of work on pseudo classically closed fields and their Galois theory, we exhibit examples of subfields of ¯Q that satisfy a universal local-global principle with respect to certain infinite families of classical closures, which may be called generalized pseudo classically closed fields. As an application, we find further evidence for the Shafarevich Conjecture. We also show that for any infinite set S0 of primes of a number field k satisfying {1}-convergence, the decomposition groups of primes in S0 generate a generalized profinite free product in Gal(k).