<p>First-principles studies of materials with strong electronic correlations introduce additional theoretical complexities and computational costs, even more so if lattice degrees of freedom also play a role. To address these challenges, we show how direct and phonon-mediated electron–electron interactions can be treated consistently. The key observation is that the GW and Fan–Migdal self-energies are topologically the same diagram, so that they can be unified using a single effective interaction, which is the sum of the two interactions. The Dyson equation for the electron Green function then yields normal-state Eliashberg equations, which can be viewed as a subset of the Hedin–Baym equations for coupled electrons and phonons [<a href="https://doi.org/10.1103/PhysRevX.13.031026" target="_blank" rel="noopener">Phys. Rev. X 13, 031026 (2023)</a>]. We solve these dynamical equations in a local approximation to calculate from first principles the temperature-dependent quasiparticle spectrum, with close attention to the low-energy window determining transport. We apply this extended dynamical Hubbard approach to the correlated metal Sr₂RuO₄, which has long served as a benchmark system for advanced electronic-structure methods. The resulting renormalization of quasiparticle bands is in qualitative agreement with experiments and results from dynamical mean-field theory. However, here the low-energy linewidth and associated resistivity within the Green–Kubo formalism in the bubble approximation are underestimated, suggesting the need for vertex corrections. This work highlights how simple dynamical formulations can provide an extensible framework for studying correlated materials, which treats electronic and vibrational excitations on equal footing.</p>