  We present the first results from SPHINX-MHD, a suite of cosmological
radiation-magnetohydrodynamics simulations designed to study the impact of
primordial magnetic fields (PMFs) on galaxy formation and the evolution of the
intergalactic medium during the epoch of reionization. The simulations are
among the first to employ on-the-fly radiation transfer and constrained
transport ideal MHD in a cosmological context to simultaneously model the
inhomogeneous process of reionization and the growth of PMFs. We run a series
of $(5{\rm Mpc})^3$ cosmological volumes, varying both the strength of the seed
magnetic field and its spectral index. We find that PMFs with a spectral index
($n_B$) and a comoving amplitude ($B_0$) that have
$n_B>-0.562\log_{10}(B_0/1{\rm n}G) - 3.35$ produce electron optical depths
($\tau_e$) that are inconsistent with CMB constraints due to the
unrealistically early collapse of low-mass dwarf galaxies. For $n_B\geq-2.9$,
our constraints are considerably tighter than the $\sim{\rm n}G$ constraints
from Planck. PMFs that do not satisfy our constraints have little impact on the
reionization history or the shape of the UV luminosity function. Likewise,
detecting changes in the Ly$\alpha$ forest due to PMFs will be challenging
because photoionisation and photoheating efficiently smooth the density field.
However, we find that the first absorption feature in the global 21cm signal is
a particularly sensitive indicator of the properties of the PMFs, even for
those that satisfy our $\tau_e$ constraint. Furthermore, strong PMFs can
increase the escape of LyC photons by up to 25% and shrink the effective radii
of galaxies by 44% which could increase the completeness fraction of galaxy
surveys. Finally, our simulations show that surveys with a magnitude limit of
${\rm M_{UV,1500{\rm A}}=-13}$ can probe the sources that provide the 50% of
photons for reionization out to $z=12$.
