Abstract The electromagnetic (EM) signal of a binary neutron star (BNS) merger depends sensitively on the total binary mass, M tot, relative to various threshold masses set by the neutron star (NS) equation of state (EOS), parameterized through the neutron star (NS) maximum mass, M TOV, and characteristic radius, R 1.6. EM observations of a BNS merger detected through its gravitational-wave (GW) emission, which are of sufficient quality to ascertain the identity of the merger remnant, can therefore constrain the values of M TOV and R 1.6, given the tight connection between M tot and the well-measured chirp mass. We elucidate the present and future landscape of EOS constraints from BNS mergers, introducing the “Multi-Messenger Matrix,” a mapping between GW and EM measurables that defines the ranges of event chirp masses that provide the most leverage on constraining the EOS. By simulating a population of BNS mergers drawn from the Galactic double NS mass distribution we show that ∼10 joint detections can constrain M TOV and R 1.6 to several percent level where systematic uncertainties may become significant. Current EOS constraints imply that most mergers will produce supramassive or hypermassive remnants, a smaller minority (possibly zero) will undergo prompt collapse, while at most only a few percent of events will form indefinitely stable NSs. In support of the envisioned program, we advocate in favor of Laser Interferometer Gravitational-Wave Observatory (LIGO)/Virgo releasing chirp mass estimates as early as possible to the scientific community, enabling observational resources to be allocated in the most efficient way to maximize the scientific gain from multi-messenger discoveries.