High-energy photons can produce electron–positron pairs upon interacting with the extragalactic background light. These pairs will in turn be deflected by the intergalactic magnetic field (IGMF), before possibly up-scattering photons of the cosmic microwave background, thereby initiating an electromagnetic cascade. The nonobservation of an excess of GeV photons and an extended halo around individual blazars due to this electromagnetic cascade can be used to constrain the properties of the IGMF. In this work, we use publicly available data of 1ES 0229+200 obtained with the Fermi Large Area Telescope and the High Energy Stereoscopic System to constrain cosmological MHD simulations of various magnetogenesis scenarios, and find that all models without a strong space-filling primordial component or overoptimistic dynamo amplifications can be excluded at the 95% confidence level. In fact, we find that the fraction of space filled by a strong IGMF has to be at least f ≳ 0.67, thus excluding most astrophysical production scenarios. Moreover, we set lower limits of B 0 > 5.1 × 10−15 G (B 0 > 1.0 × 10−14 G) for a space-filling primordial IGMF for a blazar activity time of Δt = 104 yr (Δt = 107 yr).