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The topic of this paper is achieving finite-time max-consensus in a multi-agent system that communicates over a fading wireless channel and exploits its interference property. This phenomenon corrupts the desired information when data is transmitted synchronously. In fact, each transmitted signal is attenuated by an unknown and time-varying factor (fading coefficient), then, by interference, all such attenuated signals are summed up at a receiver. Rather than combatting interference, we design a communication system that exploits it. Our strategy yields a more efficient usage of wireless resources compared to other algorithms. By simultaneously accessing this communication system, each agent obtains a weighted average of the neighbouring agents' information states. With this piece of information at hand and with a switching consensus protocol employing broadcast authorisations for agents, max-consensus can be achieved within a finite number of iterations.