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 L(tv), 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 L(tv) phase coexists with the well-known L(4) phase of spherical vesicles and a bulk L(alpha) phase. However, the defining characteristic oi the L(tv) phase is the Weissenberg rod climbing effect under shear, which results from its polymer-like entangled microstructure.