On Minkowski spacetime, the angular momentum flux through null infinity of Maxwell fields, computed using the stress-energy tensor, depends not only on the radiative degrees of freedom (d.o.f.), but also on the Coulombic parts. However, the angular momentum also can be computed using other conserved currents associated with a Killing field, such as the Noether current and the canonical current. The flux computed using these latter two currents is purely radiative. A priori, it is not clear which of these is to be considered the "true" flux of angular momentum for Maxwell fields. This situation carries over to Maxwell fields on nondynamical, asymptotically flat spacetimes for fluxes associated with the Lorentz symmetries in the asymptotic Bondi-Metzner-Sachs (BMS) algebra. We investigate this question of angular momentum flux in the full Einstein-Maxwell theory. Using the prescription of Wald and Zoupas, we compute the charges associated with any BMS symmetry on cross sections of null infinity. The change of these charges along null infinity then provides a flux. For Lorentz symmetries, Maxwell fields contribute an additional term, compared to the Wald-Zoupas charge in vacuum general relativity, to the charge on a cross section. With this additional term, the flux associated with Lorentz symmetries, e.g., the angular momentum flux, is purely determined by the radiative d.o.f. of the gravitational and Maxwell fields. In fact, the contribution to this flux by Maxwell fields is given by the radiative Noether current flux and not by the stress-energy flux.