  We study the pairing of fermions by an interaction consisting of a Hubbard
repulsion, mimicking a screened Coulomb potential, and a dynamical
phonon-mediated attraction. For such interaction, the gap equation allows even-
and odd-frequency solutions $\Delta_e$ and $\Delta_o$. We show that
odd-frequency pairing does not develop within the Eliashberg approximation due
to over-critical pair-breaking from the self-energy. When vertex corrections
are included, the pairing interaction gets stronger, and $\Delta_o$ can
develop. We argue that even in this case keeping the self-energy is still a
must as it cancels out the thermal piece in the gap equation. We further argue
that $\Delta_o$ is not affected by Hubbard repulsion and for strong repulsion
is comparable to a reduced $\Delta_e$. The resulting superconducting state is a
superposition $\Delta_e \pm i \Delta_o$, which spontaneously breaks the
time-reversal symmetry, despite that the pairing symmetry is an ordinary
$s$-wave.
