A central goal of life science has been to understand the limits of species coexistence. However, we know surprisingly little about the structure of species coexistence below such limits, and how it affects the assembly and disassembly of ecological systems. Here we introduce a novel hypergraph-based formalism that fully captures the structure of coexistence in multispecies systems. Our formalism uncovers that, below its limits, coexistence in ecological systems has ubiquitous discontinuities that we call “coexistence holes.” These coexistence holes do not occur arbitrarily but tend to obey patterns that make them predictable. We provide direct evidence showing that the biotic and abiotic constraints of empirical systems produce an over-representation of coexistence holes. By highlighting discontinuities in the form of coexistence holes, our work provides a new platform to uncover the order and structure of the assembly and disassembly of ecological systems.