Cosmic Dawn II (CoDa II) is a new, fully coupled radiation-hydrodynamics simulation of cosmic reionization and galaxy formation and their mutual impact, to redshift z < 6. With 4096^3 particles and cells in a 94 Mpc box, it is large enough to model global reionization and its feedback on galaxy formation while resolving all haloes above 10^8 M_⊙. Using the same hybrid CPU–GPU code RAMSES–CUDATON as CoDa I in Ocvirk et al. (2016), CoDa II modified and re-calibrated the subgrid star formation algorithm, making reionization end earlier, at z ≳ 6, thereby better matching the observations of intergalactic Lyman α opacity from quasar spectra and electron-scattering optical depth from cosmic microwave background fluctuations. CoDa II predicts a UV continuum luminosity function in good agreement with observations of high-z galaxies, especially at z = 6. As in CoDa I, reionization feedback suppresses star formation in haloes below ∼2 × 10^9 M_⊙, though suppression here is less severe, a possible consequence of modifying the star formation algorithm. Suppression is environment dependent, occurring earlier (later) in overdense (underdense) regions, in response to their local reionization times. Using a constrained realization of lambda cold dark matter constructed from galaxy survey data to reproduce the large-scale structure and major objects of the present-day Local Universe, CoDa II serves to model both global and local reionization. In CoDa II, the Milky Way and M31 appear as individual islands of reionization, i.e. they were not reionized by the progenitor of the Virgo cluster, or by nearby groups, or by each other.