We study a 1D model of a diffraction-mediated self-structuring instability which can occur when a Bose–Einstein condensate is illuminated by a pump laser and its reflection from a single feedback mirror. We carry out a linear stability analysis and, using numerical simulations, investigate the dynamics of the self-structuring process. Two dynamical regimes are identified: one in which the system behaves as a continuous space-time crystal oscillating between two states (one spatially uniform and one spatially periodic) and another where many condensate momentum states are involved and the condensate density develops chevrons which form and disperse quasi-periodically. We show the dependence of the pattern modulation depth and pattern formation time on pump saturation parameter and compare the simulation results with analytical expressions derived from a quantum Hamiltonian Mean Field model. The results show that this system offers a route to the first experimental realisation of the quantum Hamiltonian Mean Field model and of a continuous space-time crystal with a tunable spatial period.