A defect-induced, critical phase separation in dipolar fluids is predicted, which replaces the usual Liquid-gas transition that is driven by the isotropic aggregation of particles and is absent in dipolar fluids due to strong chaining. The coexisting phases are a dilute gas of chain ends that coexists with a high-density Liquid of chain branching points. Our model provides a unified explanation for the branched structures, the unusually Low critical temperature and density, and the consequent two-phase coexistence "islands" that were recently observed in experiment and simulation.