The observation of a massive galaxy with an extremely low dark matter content (i.e. NGC 1277) has posed questions about how such objects form and evolve in a hierarchical universe. We here report on the finding of several massive, dark matter-deficient galaxies in a set of 324 galaxy clusters theoretically modelled by means of full-physics hydrodynamical simulations. We first focus on two example galaxies selected amongst the most massive and dark matter-deficient ones. By tracing the evolution of these galaxies, we find that their lack of dark matter is a result of multiple pericentre passages. While orbiting their host halo, tidal interactions gradually strip away dark matter while preserving the stellar component. A statistical analysis of all massive satellite galaxies in the simulated clusters shows that the stellar-to-total mass ratio today is strongly influenced by the number of orbits and the distance at pericentres. Galaxies with more orbits and closer pericentres are more dark matter-deficient. Additionally, we find that massive, dark matter-deficient galaxies at the present day are either the remnants of very massive galaxies at infall or former central galaxies of infalling groups. We conclude that such massive yet dark matter-deficient galaxies exist and are natural by-products of typical cluster galaxy evolution, with no specific requirement for an exotic formation scenario.