Cooperative tasks of large-scale industrial multi-agent systems are often subject to inevitable high cost and constrains induced by inter-agent communications. To address this important issue, in this paper, we propose a novel event-triggered method for the fundamental problem—output-feedback consensus of multi-agent systems. The proposed method relies on fully distributed control and edge-based event-triggered communications. By utilizing the idea of edge-based nonmonotonic clock variables, the proposed method guarantees the existence of strictly positive inter-event times between events further than achieving Zeno-freeness, and the number of required communications is reduced. Theoretical results are verified via a numerical example.