  Several unconventional superconductors show indications of zero-energy
excitations in the superconducting state consistent with the existence of a
so-called Bogoliubov Fermi surface (BFS). In particular, FeSe doped with S
seems to acquire a nonzero density of states at zero energy at low temperatures
when doped into the tetragonal phase, consistent with a previously proposed
phenomenological theory assuming an anisotropic spin singlet pairing gap
coexisting with a nonunitary interband triplet component. Here we search for a
microscopic model that can support the coexistence of singlet pairing with
other orders, including interband nonunitary triplet pairing, and discuss
several candidates that indeed stabilize ground states with Bogoliubov Fermi
surfaces. We show that with proper choice of the coupling strength of the
various orders in our model, spontaneous breaking of $C_4$ rotational symmetry
is realized at low temperatures, in accordance with recent angle-resolved
photoemission experiments in Fe(Se,S) in the tetragonal phase.
