The Durham galform semi-analytical galaxy formation model has been shown to reproduce the observed rest-frame 1500-Å luminosity function of galaxies well over the whole redshift range z= 5–10. We show that in this model, this galaxy population also emits enough ionizing photons to reionize the Universe by redshift z= 10, assuming a modest escape fraction of 20 per cent. The bulk of the ionizing photons is produced in faint galaxies during starbursts triggered by galaxy mergers. The bursts introduce a dispersion up to ∼5 dex in galaxy-ionizing luminosity at a given halo mass. Almost 90 per cent of the ionizing photons emitted at z= 10 are from galaxies below the current observational detection limit at that redshift. Photoionization suppression of star formation in these galaxies is unlikely to affect this conclusion significantly, because the gas that fuels the starbursts has already cooled out of their host haloes. The galaxies that dominate the ionizing emissivity at z= 10 are faint, with M1500,AB∼−16, have low star formation rates, −1, and reside in haloes of mass M∼ 109 h−1 M⊙.