Topological insulators (TIs) are bulk insulators with exotic 'topologically protected' surface conducting modes. It has recently been pointed out that when stacked together, interactions between surface modes can induce diverse phases including the TI, Dirac semimetal, and Weyl semimetal. However, currently a full experimental understanding of the conditions under which topological modes interact is lacking. Here, working with multilayers of the TI Sb₂Te₃ and the band insulator GeTe, we provide experimental evidence of multiple topological modes in a single Sb₂Te₃-GeTe-Sb₂Te₃ structure. Furthermore, we show that reducing the thickness of the GeTe layer induces a phase transition from a Dirac-like phase to a gapped phase. By comparing different multilayer structures we demonstrate that this transition occurs due to the hybridisation of states associated with different TI films. Our results demonstrate that the Sb₂Te₃-GeTe system offers strong potential towards manipulating topological states as well as towards controlledly inducing various topological phases.