Recent observations have constrained the galaxy ultraviolet (UV) luminosity function up to z∼ 10. However, these observations alone allow for a wide range of reionization scenarios due to uncertainties in the abundance of faint galaxies and the escape fraction of ionizing photons. We show that requiring continuity with post-reionization (z < 6) measurements, where the Lyα forest provides a complete probe of the cosmological emissivity of ionizing photons, significantly reduces the permitted parameter space. Models that are simultaneously consistent with the measured UV luminosity function, the Thomson optical depth to the microwave background and the Lyα forest data require either (1) extrapolation of the galaxy luminosity function down to very faint UV magnitudes Mlim∼−10, corresponding roughly to the UV background suppression scale; (2) an increase of the escape fraction by a factor ≳10 from z= 4 (where the best fit is 4 per cent) to 9; or (3) more likely, a hybrid solution in which undetected galaxies contribute significantly and the escape fraction increases more modestly. Models in which star formation is strongly suppressed in low-mass, reionization-epoch haloes of mass up to Mh∼ 1010 M⊙ (e.g. owing to a metallicity dependence) are only allowed for extreme assumptions for the redshift evolution of the escape fraction. However, variants of such models in which the suppression mass is reduced (e.g. assuming an earlier or higher metallicity floor) are in better agreement with the data. Interestingly, concordance scenarios satisfying the available data predict a consistent redshift of 50 per cent ionized fraction zreion(50 per cent) ∼ 10. On the other hand, the duration of reionization is sensitive to the relative contribution of bright versus faint galaxies, with scenarios dominated by faint galaxies predicting a more extended reionization event. Scenarios relying too heavily on high-redshift dwarfs are disfavoured by kinetic Sunyaev–Zeldovich measurements, which prefer a short reionization history.