  A recent construction of polylogarithms on Riemann surfaces of arbitrary
genus in arXiv:2306.08644 is based on a flat connection assembled from
single-valued non-holomorphic integration kernels that depend on two points on
the Riemann surface. In this work, we construct and prove infinite families of
bilinear relations among these integration kernels that are necessary for the
closure of the space of higher-genus polylogarithms under integration over the
points on the surface. Our bilinear relations generalize the Fay identities
among the genus-one Kronecker-Eisenstein kernels to arbitrary genus. The
multiple-valued meromorphic kernels in the flat connection of Enriquez are
conjectured to obey higher-genus Fay identities of exactly the same form as
their single-valued non-holomorphic counterparts. We initiate the applications
of Fay identities to derive functional relations among higher-genus
polylogarithms involving either single-valued or meromorphic integration
kernels.
