  We perform Eliashberg calculations for magnon-mediated superconductivity in a
normal metal, where the electron-magnon interaction arises from interfacial
coupling to antiferromagnetic insulators. In agreement with previous studies,
we find $p$-wave pairing for large doping when the antiferromagnetic interfaces
are uncompensated, and $d$-wave pairing close to half-filling when the
antiferromagnetic interfaces are compensated. However, for the $p$-wave phase,
we find a considerable reduction in the critical temperature compared to
previous weak-coupling results, as the effective frequency cutoff on the magnon
propagator in this case is found to be much smaller than the cutoff on the
magnon spectrum. The $d$-wave phase, on the other hand, relies less on
long-wavelength magnons, leading to a larger effective cutoff on the magnon
propagator. Combined with a large density of states close to half-filling, this
might allow the $d$-wave phase to survive up to higher critical temperatures.
Based on our findings, we provide new insight into how to realize interfacially
induced magnon-mediated superconductivity in experiments.
