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 multi-frequency, on-the-fly radiation transfer and constrained transport ideal MHD in a cosmological context to simultaneously model the inhomogeneous process of reionization as well as the growth of PMFs. We run a series of $(5cMpc)3$ cosmological volumes, varying both the strength of the seed magnetic field ($B0$) and its spectral index ($nB$). We find that PMFs that have $nB>-0.562\log 10(B01nG)-3.35$ produce electron optical depths ($\tau e$) that are inconsistent with CMB constraints due to the unrealistically early collapse of low-mass dwarf galaxies. For $nB\ge -2.9$, our constraints are considerably tighter than the $\sim nG$ 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 Lya 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 sensitive indicator of the properties of the PMFs, even for those that satisfy our $\tau e$ constraint. Furthermore, strong PMFs can marginally increase the escape of LyC photons by up to 25% and shrink the effective radii of galaxies by $\sim 44\%$ which could increase the completeness fraction of galaxy surveys. Finally, our simulations show that surveys with a magnitude limit of $MUV,1500=-13$ can probe the sources that provide the majority of photons for reionization out to $z=12$.