The Lorentz-contracted electromagnetic fields of the ions in ultra-relativistic heavy-ion collisions generate intense quasi-real photon fluxes. These lead to photon-induced interactions that are observed in ultra-peripheral collisions (UPCs), such as vector meson and lepton-pair production. The high photon flux also enables the occurrence of multiple photon-induced processes in a single collision. Presented is the first measurement of the coincident production of γ γ → μ + μ − and γ + A → ρ 0 + A in UPC Pb+Pb collisions at centre-of-mass energies of 5.02 TeV and 5.36 TeV with the ATLAS detector at the Large Hadron Collider. The coincident ρ 0 is identified via its decay into a π + π − pair, with the selected π + π − sample also containing the non-resonant continuum contribution. The rate of the coincident process relative to the exclusive γ γ → μ + μ − process is measured differentially in intervals of forward event activity, quantified by the Zero Degree Calorimeters. The relative rate of coincident ρ 0 production, including non-resonant π + π − contributions, is measured to be ( 9.3 ± 0.4 ( stat . ) ± 0.2 ( syst . ) ) × 10 − 3 , summed over forward event activity. Correlations between the dimuon kinematic properties, such as its mass, and the coincident ρ 0 meson production rate, are also presented. These measurements confirm the presence of multi photon-induced processes in UPC collisions, and can provide new insight into the impact parameter dependence of photon-induced vector meson production.