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