We present the OBELISK project, a cosmological radiation-hydrodynamics simulation that follows the assembly and reionization of a protocluster progenitor during the first two billion years after the big bang, down to z = 3.5. The simulation resolves haloes down to the atomic cooling limit and tracks the contribution of different sources of ionization: stars, active galactic nuclei, and collisions. The OBELISK project is specifically designed to study the coevolution of high-redshift galaxies and quasars in an environment favouring black hole growth. In this paper, we establish the relative contribution of these two sources of radiation to reionization and their respective role in establishing and maintaining the high-redshift ionizing background. Our volume is typical of an overdense region of the Universe and displays star formation rate and black hole accretion rate densities similar to those of high-redshift protoclusters. We find that hydrogen reionization happens inside-out, is completed by z similar to 6 in our overdensity, and is predominantly driven by galaxies, while accreting black holes only play a role at z similar to 4.