This work considers a network of autonomous agents that exploit the superposition property of the wireless channel for achieving max-consensus under communication constraints. In technical applications, superposition (or interference) is traditionally avoided by physically separating transmissions from different agents, e.g., by employing time-division multiple access or frequency-division multiple access protocols. In contrast, Over-the-Air computation relies on simultaneous transmissions of signals in the same frequency band to boost performance. This article extends the results presented by Molinari et al. 2021 in a significant way: Rather than requiring a new and hardly accessible technology that allows simultaneous receiving and transmission of signals (full-duplex), we present a solution based on cheap and ubiquitous half-duplex technology. However, half-duplex technology only allows agents to either receive or transmit at a given time, hence requiring agents to be grouped into clusters. The proposed method, first, enables each cluster to achieve max-consensus within itself. Then, agents belonging to more than one cluster spread the information across different clusters that are, in general, not synchronized. The proposed max-consensus protocol can deal with asynchronism and allows the network to achieve max-consensus within a finite number of steps. Simulations confirm our theoretical results.