We investigate the spatial, kinematic and chemical properties of globular cluster systems formed in merging and interacting galaxies using N-body-smoothed particle hydrodynamics (SPH) simulations. Although we cannot resolve individual clusters in our simulation, we assume that they form in collapsing molecular clouds when the local external gas pressure exceeds 105kB (where kB is the Boltzmann constant). Several simulations are carried out for a range of initial conditions and galaxy mass ratios. The input model spirals are given a halo globular cluster system similar to those observed for the Milky Way and M31. Gravitational tidal effects during galaxy merging and interaction lead to a dramatic increase in gas pressure, which exceeds our threshold and hence triggers new globular cluster formation. We investigate the properties of the globular cluster system in the remnant galaxy, such as the number density, the specific frequency, kinematic properties and the metallicity distribution. Different orbital conditions and mass ratios give rise to a range in globular cluster properties, particularly for the interaction models. Our key results are the following: the newly formed metal-rich clusters are concentrated at the centre of the merger remnant elliptical, whereas the metal-poor ones are distributed to the outer parts because of strong angular momentum transfer. The dissipative merging of present-day spirals, including chemical evolution, results in metal-rich clusters with a mean metallicity that is super-solar, i.e. much higher than is observed in elliptical galaxies. If elliptical galaxies form by dissipative major mergers, then they must do so at very early epochs when their discs contained low-metallicity gas. Our simulations show that the specific frequency can be increased in a dissipative major merger. However, when this occurs it results in a ratio of metal-poor to metal-rich clusters that is less than one, contrary to the ratio observed in many elliptical galaxies.