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 (IGM) during the epoch of reionization. The simulations are among the first to employ multifrequency, 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 primordial magnetic fields. We run a series of |$(5\, \text{cMpc})^3$| cosmological volumes, varying both the strength of the seed magnetic field and its spectral index. We find that PMFs with a spectral index () and a comoving amplitude (B_0) that have |$\scriptstyle{n_B \gt -0.562\log _{10}\left(\frac{B_0}{1{\rm n}G}\right) - 3.35}$| produce electron optical depths (τ_e) that are inconsistent with CMB constraints due to the unrealistically early collapse of low-mass dwarf galaxies. For n_B ≥ −2.9, our constraints are considerably tighter than the ∼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 Ly α forest due to PMFs will be challenging because photoionization and photoheating efficiently smooth the density field. However, we find that the first absorption feature in the global 21-cm signal is a particularly sensitive indicator of the properties of the PMFs, even for those that satisfy our τ_e constraint. Furthermore, strong PMFs can marginally increase the escape of LyC photons by up to 25 per cent and shrink the effective radii of galaxies by |$\sim 44{{\ \rm per\ cent}}$| which could increase the completeness fraction of galaxy surveys. Finally, our simulations show that surveys with a magnitude limit of |${\rm \mathit{ M}_{UV,1500\,\mathring{\rm A}}=-13}$| can probe the sources that provide the majority of photons for reionization out to z = 12.