Recent observations suggest that hydrogen reionization ends late ($z\simeq 6$) and proceeds quickly. We present here a new model of the meta-galactic UV/X-ray background (UVB) that is consistent with this. It adopts the most recent determinations of the ionizing emissivity due to stars and AGN, as well as of the HI absorber column density distribution. Another major improvement is a new treatment of the intergalactic medium (IGM) opacity for ionizing photons that is able to consistently capture the transition from a neutral to an ionized IGM. Previous synthesis models of the UVB, when used in simulations, yield reionization and thermal histories that are inconsistent with the assumed ionizing emissivities. With our new treatment, this discrepancy is fully resolved. In our fiducial model, galaxies leaking $\lesssim 18\%$ of their Lyman continuum emission drive HI reionization, while AGN drive HeII reionization (completing at $z\simeq 6.2$ and $2.8$ respectively). Due to the limited time available for cooling between HI and HeII reionization, higher IGM temperatures are predicted for late reionization scenarios. In our fiducial model, the predicted temperatures agree well with observational constraints at $z\lesssim 4$, while being slightly high compared to (somewhat uncertain) data above that. Models with a larger contribution of AGN are instead disfavoured by the temperature data, as well as by measurements of the HI and HeII Lyman-$\alpha$ forest opacities. We also present "equivalent-equilibrium" ionization/heating rates that mimic our fiducial UVB model for use in simulation codes that assume ionization equilibrium.