Scanning tunneling microscopy (STM) and density-functional theory are used to reexamine the structure of the renowned p(4 x 4)-O/Ag(111) surface oxide. The accepted structural model [C. I. Carlisle, Phys. Rev. Lett. 84, 3899 (2000)10.1103/PhysRevLett.84.3899] is incompatible with the enhanced resolution of the current STM measurements. An "Ag6 model" is proposed that is more stable than its predecessor and accounts for the coexistence of the p(4 x 4) and a novel c(3 x 5log3)rect phase. This coexistence is an indication of the dynamic complexity of the system that until now has not been appreciated.