Subsurface bilayers are found to exist at the air-water interface of spontaneously forming vesicular solutions of sodium bis(2-ethylhexyl) sulfosuccinate, (AOT). The bilayers produce long-range repulsive forces that increase the stability of foam and emulsion films formed from these solutions. In addition, during film formation and adhesion of two interfaces possessing subsurface bilayers, tubular networks of wrinkled bilayers become trapped within the film. This phenomenon is analogous to pocket formation in lipid membrane multilayers seen during a binding transition, and the resulting networks slowly evolve through processes that mimic two-dimensional foam coarsening.