Recent endeavors to combine the desirable energy-absorption characteristics of stochastic foams with the comparatively high strengths of pyramidal lattices have shown promise for creating composites that outperform their constituents alone under compressive loading. Herein we employ numerical and analytical models to identify both the mechanisms by which synergistic behavior is obtained in such composites and the constituent mass fractions that yield maximum benefits. We find that the loading boundary conditions play a crucial role. When, for instance, composites are loaded between plates that are well bonded to the composites, their specific strengths invariably exceed those predicted by a rule-of-mixtures; however, these strengths can always be improved through an optimized lattice of equivalent mass. In contrast, when the composites are loaded between frictionless plates, their specific strengths exceed not only rule-of-mixtures predictions but, in many cases, also that of any mass-equivalent pyramidal lattice alone subject to the same (frictionless) conditions. The origin of this behavior is found to arise from foam-stabilization of lattice bending and splaying: deformation modes that govern strength in the absence of foam. In essence, the foam causes a transition from bend-dominated to stretch-dominated behavior in the lattice.