We present experimental data on the phase diagram and the properties of the phases for a ternary surfactant system of a zwitterionic single-chain hydrocarbon surfactant (tetradecyldimethylaminoxide, TDMAO) and a perfluoro cosurfactant (PFC) (1,1-H-dihydroperfluorooctanol) in water. The phases were studied for the two conditions in which 10% of the zwitterionic surfactant was ionically charged by HCl (case a) and in which 10 mM NaCl was added to the surfactant solutions (case b). For both cases we observed the same sequence of phases as for hydrocarbon surfactant/cosurfactant mixtures, namely, L-1, L-1/L-alpha, and L-alpha. An L-3 or sponge phase, which is normally observed after the L-alpha phase, was not observed with the perfluoro cosurfactant. In case a the L-alpha phase for a 100 mM surfactant concentration is stable over a wide cosurfactant/surfactant ratio from 0.4 to 1. The L-alpha phase is highly viscoelastic and has a yield stress value. FF-TEM micrographs show that the phase consists of two types of vesicles : small unilamellar ones (SUV) and large multilamellar ones (LMV). It is conceivable that the two types of vesicles have different compositions. Theoretical considerations make it likely that the SW have a lower cosurfactant/surfactant ratio and a higher charge density than the LMV. The charged L-alpha phase can be diluted to concentrations as low as 10 mM without any obvious signs of macroscopic phase separation. The viscoelastic properties disappear for concentrations c < 60 mM. For case b, the L-alpha phase of 100 mM surfactant separates macroscopically into two phases, both of which contain vesicles. The lower phase has a higher cosurfactant/surfactant ratio than the upper phase. These results indicate that the uncharged bilayers of hydrocarbon surfactants and perfluoro cosurfactants have a demixing gap, and the consequence is a perfluoro-rich phase and a perfluoro-poor phase. When the bilayers are charged (case a), the macroscopic demixing disappears while the local demixing probably remains.