Primordial Magnetogenesis in the Two-Higgs-doublet Model

$\gamma$-ray emission of blazars infer the presence of large-scale magnetic fields in the intergalactic medium, but their origin remains a mystery. Using recent data from MAGIC, H.E.S.S. and $\textit{Fermi}$-LAT, we investigate whether the large-scale magnetic fields in the intergalactic medium could have been generated by a first-order electroweak phase transition in the two-Higgs-doublet model (2HDM). We study two representative scenarios where we vary the initial conditions of the magnetic field and the plasma, assuming either a primordial magnetic field with maximal magnetic helicity or a primordial magnetic field with negligible magnetic helicity in a plasma with kinetic helicity. By considering a primordial magnetic field with maximal helicity and applying the conservative constraints derived from MAGIC and $\textit{Fermi}$-LAT data, we demonstrate that a first-order electroweak phase transition within the 2HDM may account for the observed intergalactic magnetic fields in the case of the strongest transitions. We show that this parameter space also predicts strong gravitational wave signals in the reach of space-based detectors such as LISA, providing a striking multi-messenger signal of the 2HDM.