Abstract We use the IllustrisTNG cosmological hydrodynamical simulation to study the formation of ultra-diffuse galaxies (UDGs) in galaxy clusters. We supplement the simulations with a realistic mass-size relation for galaxies at the time of infall into the cluster, as well as an analytical model to describe the tidally-induced evolution of their stellar mass, velocity dispersion and size. The model assumes ‘cuspy’ NFW halos and, contrary to recent claims, has no difficulty reproducing the observed number of UDGs in clusters. Our results further suggest that the UDG population consists of a mixture of ‘normal’ low surface brightness galaxies such as those found in the field (‘born’ UDGs, or B-UDGs), as well as a distinct population that owe their large size and low surface brightness to the effects of cluster tides (‘tidal’, or T-UDGs). The simulations indicate that T-UDGs entered the cluster earlier and should be more prevalent than B-UDGs near the cluster centres. T-UDGs should also have, at given stellar mass, lower velocity dispersion, higher metallicities, and lower dark matter content than B-UDGs. Our results suggest that systems like DF-44 are consistent with having been born as UDGs, while others such as DF2, DF4 and VLSB-D are possibly extreme T-UDG examples.