  Binary neutron star mergers (NSMs) have been confirmed as one source of the
heaviest observable elements made by the rapid neutron-capture (r-) process.
However, modeling NSM outflows -- from the total ejecta masses to their
elemental yields -- depends on the unknown nuclear equation of state (EOS) that
governs neutron-star structure. In this work, we derive a phenomenological EOS
by assuming that NSMs are the dominant sources of the heavy element material in
metal-poor stars with r-process abundance patterns. We start with a population
synthesis model to obtain a population of merging neutron star binaries and
calculate their EOS-dependent elemental yields. Under the assumption that these
mergers were responsible for the majority of r-process elements in the
metal-poor stars, we find parameters representing the EOS for which the
theoretical NSM yields reproduce the derived abundances from observations of
metal-poor stars. For our proof-of-concept assumptions, we find an EOS that is
slightly softer than, but still in agreement with, current constraints, e.g.,
by the Neutron Star Interior Composition Explorer, with $R_{1.4}=12.25\pm
0.03$~km and $M_{\textrm TOV}$ of $2.17\pm 0.03$~M$_\odot$(statistical
uncertainties, neglecting modeling systematics).
