  We determine the baryon spectrum of 1 + 1 + 1-flavor QCD in the presence of
strong background magnetic fields using lattice simulations at physical quark
masses for the first time. Our results show a splitting within multiplets
according to the electric charge of the baryons and reveal, in particular, a
reduction of the nucleon masses for strong magnetic fields. This
first-principles input is used to define constituent quark masses and is
employed to set the free parameters of the Polyakov loop-extended
Nambu-Jona-Lasinio (PNJL) model in a magnetic field-dependent manner. The so
constructed model is shown to exhibit inverse magnetic catalysis at high
temperatures and a reduction of the transition temperature as the magnetic
field grows - in line with non-perturbative lattice results. This is contrary
to the naive variant of this model, which gives incorrect results for this
fundamental phase diagram. Our findings demonstrate that the magnetic field
dependence of the PNJL model can be reconciled with the lattice findings in a
systematic way, employing solely zero-temperature first-principles input.
