Increasing evidence suggests that cosmological sheets, filaments, and voids may be substantially magnetized today. The origin of magnetic fields in the intergalactic medium (IGM) is, however, currently uncertain. It seems well known that non-standard extensions to the physics of the standard model can provide mechanisms susceptible of magnetizing the universe at large. Perhaps less well known is the fact that standard, classical physics of matter--radiation interactions actually possesses the same potential. We discuss a magnetogenesis mechanism based on the exchange of momentum between hard photons and electrons in an inhomogeneous IGM. Operating in the neighborhood of ionizing sources during the epoch of reionization, this mechanism is capable of generating magnetic seeds of relevant strengths over scales comparable to the distance between ionizing sources. In addition, summing up the contributions of all ionizing sources and taking into account the distribution of gas inhomogeneities, we show that this mechanism leaves the IGM, at the end of reionization, with a level of magnetization that might account, when amplification mechanisms take over, for the magnetic fields strengths in the current cosmic web.