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We introduce a new self-consistent model of galaxy evolution and reionization, ASTRAEUS (semi-numerical rAdiative tranSfer coupling of galaxy formaTion and Reionization in N-body dArk mattEr simUlationS), which couples a state-of-the-art N-body simulation with the semi-analytical galaxy evolution DELPHI and the semi-numerical reionization scheme CIFOG. ASTRAEUS includes all the key processes of galaxy formation and evolution (including accretion, mergers, supernova and radiative feedback) and follows the time and spatial evolution of the ionized regions in the intergalactic medium (IGM). Importantly, it explores different radiative feedback models that cover the physically plausible parameter space, ranging from a weak and delayed to a strong and immediate reduction of gas mass available for star formation. From our simulation suite that covers the different radiative feedback prescriptions and ionization topologies, we find that radiative feedback continuously reduces star formation in galaxies with $M_h<10^{9.5}M_{\odot}$ upon local reionization; larger mass halos are unaffected even for the strongest and immediate radiative feedback cases during reionization. For this reason, the ionization topologies of different radiative feedback scenarios differ only on scales smaller than $1-2$Mpc, and significant deviations are only found when physical parameters (e.g. the escape fraction of ionizing photons) are altered based on galactic properties. Finally, we find observables (the ultra-violet luminosity function, stellar mass function, reionization histories and ionization topologies) are hardly affected by the choice of the used stellar population synthesis models that either model single stars or binaries.