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 TDMA (Time Division Multiple Access) or FDMA (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 paper extends the results presented in Molinari et al. [14] 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.