A fascinating inverse problem that has been receiving considerable attention is the construction of realizations of random two-phase heterogeneous media with a target two-point correlation function. However, not every hypothetical two-point correlation function corresponds to a realizable two-phase medium. Here, we collect all of the known necessary conditions on the two-point correlation functions scattered throughout a diverse literature and derive a new, but simple, positivity condition. We apply the necessary conditions to test the realizability of certain classes of proposed correlation functions.