Magnetic fields are ubiquitous in the Universe. They seem to be present at virtually all scales and all epochs. Yet, whether the fields on cosmological scales are of astrophysical or cosmological origin remains an open major problem. Here we focus on an astrophysical mechanism based on the photoionization of the intergalactic medium during the Epoch of Reionization. Building upon previous studies that depicted the physical mechanism around isolated sources of ionization, we present here an analytic model to estimate the level at which this mechanism contributed to the magnetization of the whole Universe, thanks to the distribution of sources, before and alongside early luminous structure formation. This model suggests that the Universe may be globally magnetized to the order of, at least, a few 10^−20 G comoving (i.e. several 10^−18 G during the Epoch of Reionization) by this mechanism, prior to any amplification process.