The formation of biomolecular condensates through liquid-liquid phase separation from proteins and nucleic acids is emerging as a spatial organisational principle used by living cells. Many such biomolecular condensates are not, however, homogeneous fluids, but contain an internal structure consisting of distinct sub-compartments with different compositions. In many instances, such compartments inside the condensate are depleted in the biopolymers that make up the condensate. Here, we describe that this multiphase structure arises from a kinetically arrested phase transition. The combination of a change in composition coupled with a slow response to this change can lead to the spontaneous formation of multiple emulsions consisting of several inner cores within a polymer-rich middle phase. In the case of liquid-like biomolecular condensates, the slow diffusion of biopolymers causes nucleation of biopolymer-poor liquid inside of the condensate to achieve the new equilibrium composition. This framework shows that multiphase condensates can be a result of kinetic trapping, rather than thermodynamic stability, and provides and avenue to understand and control the internal structure of condensates in vitro and in vivo .