Superconductors are characterized by the formation of an electronic condensate of pairs, which form in spite of their Coulombic repulsion. The Eliashberg theory deals with this apparent contradiction by considering the dynamical properties of the interactions.   The pairing instability arises from pair correlations that exploit the interaction's frequency dependence to extract attractive "channels" while avoiding repulsive ones. However, the role of Coulomb repulsion in the properties of the resulting superconducting state remains uncertain, in part because deriving the phenomenological Ginzburg-Landau theory for such a superconductor from a microscopic model that incorporates repulsion remains elusive. We present a formalism that addresses this challenge by applying the standard Hubbard-Stratonivich transformation to an interaction that we first decompose into attractive and repulsive channels. This leads to a complex action governed by a  saddle point that is shifted from the original field-integration manifold into a generalized complex one. Using a gradient-descent method, we obtain a numerical solver that finds the solution to Eliashberg equations efficiently and impartially. We then describe how we account for fluctuations around this complex saddle point and apply it to compute the upper critical field near the superconducting transition temperature.