Interacting galaxies are well known for their high star formation rates and rich star cluster populations, but it is also recognized that the rapidly changing tidal field can efficiently destroy clusters. We use numerical simulations of merging disc galaxies to investigate which mechanism dominates. The simulations include a model for the formation and evolution of the entire star cluster population, accounting for the evaporation of clusters due to two-body relaxation and tidal shocks. We find that the dynamical heating of stellar clusters by tidal shocks is about an order of magnitude higher in interacting galaxies than in isolated galaxies. This is driven by the increased gas density and is sufficient to destroy star clusters at a higher rate than new clusters are formed: the total number of stellar clusters in the merger remnant is 2–50 per cent of the amount in the progenitor discs, with low-mass clusters being disrupted preferentially. By adopting observationally motivated selection criteria, we find that the observed surplus of star clusters in nearby merging galaxies with respect to isolated systems is caused by the observational bias to detect young, massive clusters, and marks a transient phase in galaxy evolution. We provide a general expression for the survival fraction of clusters, which increases with the gas depletion time-scale, reflecting that both the formation and the destruction of clusters are driven by the growth of the gas density. Due to the preferential disruption of low-mass clusters, the mass distribution of the surviving star clusters in a merger remnant develops a peak at a mass of about 103 M⊙, which evolves to higher masses at a rate of 0.3–0.4 dex Gyr−1. Briefly after a merger, the peak mass depends weakly on the galactocentric radius, but this correlation disappears as the system ages due to the destruction of clusters on eccentric orbits. We discuss the similarities between the cluster populations of the simulated merger remnants and (young) globular cluster systems. Our results suggest that the combination of cluster formation and destruction should be widespread in the dense star-forming environments at high redshifts, which could provide a natural origin to present-day globular cluster systems.