Galactic centers (GCs) are very dynamically active environments, often harboring a nuclear star cluster and supermassive black hole at their cores. Binaries in these environments are subject to strong tidal fields that can efficiently torque its orbit, exciting near-unity eccentricities that ultimately lead to their merger. In turn, frequent close interactions with passing stars impulsively perturb the orbit of the binary, generally softening their orbit until their evaporation, potentially hindering the role of tides to drive these mergers. In this work, we study the evolution of compact object binaries in the GC and their merger rates, focusing for the first time on the combined effect of the cluster's tidal field and flyby interactions. We find a significant synergy between both processes, where merger rates increase by a factor of similar to 10-30 compared to models in which only flybys or tides are taken into account. This synergy is a consequence of the persistent tides-driven eccentricity excitation that is enhanced by the gradual diffusion of j z driven by flybys. The merger efficiency peaks when the diffusion rate is similar to 10-100 slower than the tides-driven torquing. Added to this synergy, we also find that the gradual softening of the binary can lift the relativistic quenching of initially tight binaries, otherwise unable to reach extreme eccentricities, and thus expanding the available phase space for mergers. Cumulatively, we conclude that despite the gradual softening of binaries due to flybys, these greatly enhance their merger rates in GCs by promoting the tidal-field-driven eccentricity excitation.