Humans regularly engage in efficient communicative conversations, which serve to socially align individuals¹. In conversations, we take fast-paced turns using a human-universal structure of deploying and receiving signals which shows consistent timing across cultures². We report here that chimpanzees also engage in rapid signal-to-signal turn-taking during face-to-face gestural exchanges with a similar average latency between turns to that of human conversation. This correspondence between human and chimpanzee face-to-face communication points to shared underlying rules in communication. These structures could be derived from shared ancestral mechanisms or convergent strategies that enhance coordinated interactions or manage competition for communicative ‘space'.