We establish the existence of a weak solution for a strongly coupled, nonlinear Stokes–Maxwell system, originally proposed by Nika and Vernescu (Z Angew Math Phys71(1):1–19, 2020) in the three-dimensional setting. The model effectively couplesthe Stokes equation with the quasi-static Maxwell’s equations through the Lorentzforce and the Maxwell stress tensor. The proof of existence is premised on: (i) theaugmented variational formulation of Maxwell’s equations, (ii) the definition of a newfunction space for the magnetic induction and the verification of a Poincar’e-typeinequality, and (iii) the deployment of the Altman–Shinbrot fixed point theorem whenthe magnetic Reynolds number, Rm, is small.