Far out-of-equilibrium many-body quantum dynamics in isolated systems necessarily generates interferences beyond an Ehrenfest timescale, where quantum and classical expectation values diverge. Of great recent interest is the role these interferences play in the spreading of quantum information across the many degrees of freedom, i.e., scrambling. Ultracold atomic gases provide a promising setting to explore these phenomena. Theoretically speaking, the heavily-relied-upon truncated Wigner approximation leaves out these interferences. We develop a semiclassical theory which bridges classical and quantum concepts in many-body bosonic systems and properly incorporates such missing quantum effects. For mesoscopically populated Bose-Hubbard systems, it is shown that this theory captures post-Ehrenfest quantum interference phenomena very accurately, and contains relevant phase information to perform many-body spectroscopy with high precision.