(Abridged) Interacting galaxies are well-known for their high star formation rates and rich star cluster populations, but the rapidly changing tidal field can also 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 dynamical disruption of the entire star cluster population. We find that the dynamical heating of 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 clusters in the merger remnant is 2-50% 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 is caused by the bias to detect young, massive clusters. We provide a general expression for the survival fraction of clusters, which increases with the gas depletion time-scale. 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 10^3 Msun, which evolves to higher masses at a rate of 0.3-0.4 dex per Gyr. The peak mass initially depends weakly on the galactocentric radius, but this correlation disappears as the system ages. 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.
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