We investigate the effect of primordial helium on hydrogen reionization using a hydrodynamical simulation combined with the cosmological radiative transfer code crash. The radiative transfer simulations are performed in a 35.12 h−1 comoving Mpc box using a variety of assumptions for the amplitude and power-law extreme-ultraviolet (EUV) spectral index of the ionizing emissivity at z > 6. We use an empirically motivated prescription for ionizing sources which, by design, ensures all of the models are consistent with constraints on the Thomson scattering optical depth and the metagalactic hydrogen photoionization rate at z∼ 6. The inclusion of helium slightly delays reionization due to the small number of ionizing photons which reionize neutral helium instead of hydrogen. However, helium has a significant impact on the thermal state of the intergalactic medium (IGM) during hydrogen reionization. Models with a soft EUV spectral index, α= 3, produce IGM temperatures at the mean density at z∼ 6, T0≃ 10 500 K, which are ∼20 per cent higher compared to models in which helium photoheating is excluded. Harder EUV indices produce even larger IGM temperature boosts by the end of hydrogen reionization. A comparison of these simulations to recent observational estimates of the IGM temperature at z∼ 5–6 suggests that hydrogen reionization was primarily driven by Population II stellar sources with a soft EUV index, . We also find that faint, as yet undetected galaxies, characterized by a luminosity function with a steepening faint-end slope (αLF≤−2) and an increasing Lyman continuum escape fraction (fesc∼ 0.5), are required to reproduce the ionizing emissivity used in our simulations at z > 6. Finally, we note there is some tension between recent observational constraints which indicate the IGM is >10 per cent neutral by volume z∼ 7, and estimates of the ionizing emissivity at z= 6 which indicate only 1–3 ionizing photons are emitted per hydrogen atom over a Hubble time at z= 6. This tension may be alleviated by either a lower neutral fraction at z∼ 7 or an IGM which still remains a few per cent neutral by volume at z= 6.