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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. Using this understanding we develop an improved numerical
solver that outperforms standard techniques to solve the non-linear equations.
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.