Thin films of Ag(111) with two-dimensional crystallinity of large lateral coherence grow on Ge(111), free of in-plane registry with the underlying substrate. Ag s-p electrons forming two-dimensional quantum well states scatter coherently at the buried interface potential, resulting in an unexpected set of new quasiparticle states, as observed by angle-resolved photoemission. These new features originate from interactions among Ag quantum well bands, gaining a momentum equivalent to a reciprocal vector of the substrate lattice.