An equilibrium phase belonging to the family of bilayer liposomes in ternary mixtures of dimyristoylphosphatidylcholine (DMPC), water, and geraniol (a biological alcohol derived from oil-soluble vitamins that acts as a cosurfactant) has been identified. Electron and optical microscopy reveal the phase, labeled Ltv, to be composed of highly entangled tubular vesicles. In situ x-ray diffraction confirms that the tubule walls are multilamellar with the lipids in the chain-melted state. Macroscopic observations show that the Ltvphase coexists with the well-known L4 phase of spherical vesicles and a bulk Lα phase. However, the defining characteristic of the Ltv phase is the Weissenberg rod climbing effect under shear, which results from its polymer-like entangled microstructure.