Modeling cooperative dynamics using networks of phase oscillators is common practice for a wide spectrum of biological and technological networks, ranging from neuronal populations to power grids. In this paper we study the emergence of stable clusters of synchrony with complex intercluster dynamics in a three-population network of identical Kuramoto oscillators with inertia. The populations have different sizes and can split into clusters where the oscillators synchronize within a cluster, but notably, there is a phase shift between the dynamics of the clusters. We extend our previous results on the bistability of synchronized clusters in a two-population network [I. V. Belykh et al., Chaos 26, 094822 (2016)CHAOEH1054-150010.1063/1.4961435] and demonstrate that the addition of a third population can induce chaotic intercluster dynamics. This effect can be captured by the old adage "two is company, three is a crowd," which suggests that the delicate dynamics of a romantic relationship may be destabilized by the addition of a third party, leading to chaos. Through rigorous analysis and numerics, we demonstrate that the intercluster phase shifts can stably coexist and exhibit different forms of chaotic behavior, including oscillatory, rotatory, and mixed-mode oscillations. We also discuss the implications of our stability results for predicting the emergence of chimeras and solitary states.