We present the first detailed structure formation and radiative transfer simulations of the reionization history of our cosmic neighbourhood. To this end, we follow the formation of the Local Group of galaxies and nearby clusters by means of constrained simulations, which use the available observational constraints to construct a representation of those structures which reproduces their actual positions and properties at the present time. We find that the reionization history of the Local Group is strongly dependent on the assumed photon production efficiencies of the ionizing sources, which are still poorly constrained. If sources are relatively efficient, i.e. the process is 'photon-rich', the Local Group is primarily ionized externally by the nearby clusters. Alternatively, if the sources are inefficient, i.e. reionization is 'photon-poor' the Local Group evolves largely isolated and reionizes itself. The mode of reionization, external versus internal, has important implications for the evolution of our neighbourhood, in terms of e.g. its satellite galaxy populations and primordial stellar populations. This therefore provides an important avenue for understanding the young universe by detailed studies of our nearby structures.