In these proceedings we present preliminary results for the masses of the proton, neutron and $\Omega^-$ baryons obtained from QCD+QED lattice simulations performed with four dynamical quarks using C$^*$ boundary conditions.
These results are part of the ongoing effort of the RC${}^*$ collaboration discussed in the companion proceedings, and have been obtained on a single ensemble in which the renormalised electromagnetic coupling is $\alpha_{\text{em}}\sim 0.04$, the physical volume is $L\sim 1.7$ fm and the masses of the four dynamical quarks have been tuned at the $U$--spin symmetric point $m_d=m_s$. We demonstrate on this unphysical ensemble that baryon masses can be calculated with satisfactory precision when including QED without the need for gauge--fixing and perturbation theory. This makes us confident in the effectiveness of the strategy presented here also in the case of simulations closer to the physical point.