In many biomolecular solutions, multivalent interactions that promote complexation lead to phase separation at high concentrations. Here we study a model biomolecular system that exhibits the opposite behavior: Self-assembly into multisubunit complexes inhibits phase separation, even though the same intermolecular interactions underlie both processes. We use microfluidics-based experiments to show that DNA nanostars, which are known to phase separate at low temperatures, can also phase separate upon disassembly at high temperatures and high concentrations of divalent cations. We introduce a theoretical model that quantitatively reproduces this novel re-entrant phase behavior, providing a unified view of the competition between self-assembly and phase separation in biomolecular solutions.