LLMs are used predominantly in synchronous communication, where a human user and a model communicate in alternating turns. In contrast, many real-world settings are asynchronous. For example, in group chats, online team meetings, or social games, there is no inherent notion of turns. In this work, we develop an adaptive asynchronous LLM agent consisting of two modules: a generator that decides what to say, and a scheduler that decides when to say it. To evaluate our agent, we collect a unique dataset of online Mafia games, where our agent plays with human participants. Overall, our agent performs on par with human players, both in game performance metrics and in its ability to blend in with the other human players. Our analysis shows that the agent's behavior in deciding when to speak closely mirrors human patterns, although differences emerge in message content. We make all of our code and data publicly available. This work paves the way for integration of LLMs into realistic human group settings, from assistance in team discussions to educational and professional environments where complex social dynamics must be navigated.