For a metric space (X,d), we consider the so-called Lipschitz realcompactification of X, denoted by H(Lipd(X)). In this note we give a result concerning the equalityH(Lipd+ρ(X×Y))=H(Lipd(X))×H(Lipρ(Y)) for the product of the two metric spaces (X,d) and (Y,ρ). More precisely, we prove that such equality holds if and only if H(Lipd(X))=X˜ or H(Lipρ(Y))=Y˜, where X˜ and Y˜ denote the completion of X and Y respectively, or equivalently, if and only if the Lipschitz realcompactification of one of the factors X or Y is as simple as possible. We also point out that our result is, in fact, a true generalization of a known theorem by Woods about the Samuel compactification of the product of two metric spaces.