In this work, we computed the equation of state of dense QCD in the presence of background magnetic fields using lattice QCD simulations at imaginary baryon chemical potential. Our simulations include 2+1+1 flavors of stout-smeared staggered fermions with masses at the physical point and a tree-level Symanzik-improved gauge action. Using several expansion schemes, we tuned our simulation parameters such that the equation of state satisfies strangeness neutrality and isospin asymmetry constraints, which are relevant to the phenomenology of heavy-ion collisions. Our results suggest a strong change in the equation of state due to the magnetic field, in particular, around the crossover temperature. A continuum extrapolation of our data is still needed for future applications of our equation of state to heavy-ion-collision phenomenology.