  We investigate the effect of the Biermann battery during the Epoch of
Reionization (EoR) using cosmological Adaptive Mesh Refinement simulations
within the framework of the SPHINX project. We develop a novel numerical
technique to solve for the Biermann battery term in the Constrained Transport
method, preserving both the zero divergence of the magnetic field and the
absence of Biermann battery for isothermal flows. The structure-preserving
nature of our numerical method turns out to be very important to minimise
numerical errors during validation tests of the propagation of a Str\"omgren
sphere and of a Sedov blast wave. We then use this new method to model the
evolution of a 2.5 and 5 co-moving Mpc cosmological box with a state-of-the-art
galaxy formation model within the RAMSES code. Contrary to previous findings,
we show that three different Biermann battery channels emerge: the first one is
associated with linear perturbations before the EoR, the second one is the
classical Biermann battery associated with reionization fronts during the EoR,
and the third one is associated with strong, supernova-driven outflows. While
the two former channels generate spontaneously volume-filling magnetic fields
with a strength on the order or below $10^{-20}$ G, the latter, owing to the
higher plasma temperature and a marginally-resolved turbulent dynamo, reaches a
field strength as high as $10^{-18}$ G in the intergalactic medium around
massive haloes.
