  A superconductor emerges as a condensate of electron pairs, which bind
despite their strong Coulomb repulsion. Eliashberg's theory elucidates the
mechanisms enabling them to overcome this repulsion and predicts the transition
temperature and pairing correlations. However, a comprehensive understanding of
how repulsion impacts the phenomenology of the resulting superconductor remains
elusive. We present a formalism that addresses this challenge by applying the
Hubbard-Stratonovich transformation to an interaction including instantaneous
repulsion and retarded attraction. We first decompose the interaction into
frequency scattering channels and then integrate out the fermions. The
resulting bosonic action is complex and the saddle point corresponding to
Eliashberg's equations generally extends into the complex plane and away from
the physical axis. We numerically determine this saddle point using the
gradient descent method, which is particularly well-suited for the case of
strong repulsion. We then turn to consider fluctuations around this complex
saddle point. The matrix controlling fluctuations about the saddle point is
found to be a non-Hermitian symmetric matrix, which generally suffers from
exceptional points that are tuned by different parameters. These exceptional
points may influence the thermodynamics of the superconductor. For example,
within the quadratic approximation the upper critical field sharply peaks at a
critical value of the repulsion strength related to an exceptional point
appearing at $T_c$. Our work facilitates the mapping between microscopic and
phenomenological theories of superconductivity, particularly in the presence of
strong repulsion. It has the potential to enhance the accuracy of theoretical
predictions for experiments in systems where the pairing mechanism is unknown.
