Interlayer excitons, comprising an electron in one material bound by Coulomb attraction to a hole in an adjacent material, are composite bosons that can assume a variety of many-body phases. The phase diagram of the bosonic system is largely determined by the dispersion relation of the bosons, which itself arises as a combination of the dispersion relations of the electron and hole separately. Here I show that in situations where either the electron or the hole has a non-monotonic, "Mexican hat-shaped" dispersion relation, the exciton dispersion relation can have a range of qualitatively different forms, each corresponding to a different many-body phase at low temperature. This diversity suggests a novel platform for continuously tuning between different quantum phases using an external field.