Monitoring and maintaining communications in a multilayer ad hoc wireless network requires a stable communication overlay. Such monitoring should be resilient, avoiding reliance on a single (or a few) layers, and energy-aware, not being vulnerable to the exhaustion of available energy in a single network element. Thus, there is possibly a trade-off between overlay size, interlayer connectivity, and the energy distribution within the overlay. In this paper, we present three distributed energy-aware multilayer connected edge dominating set algorithms, show how they manage such trade-offs in practical scenarios, and show that CCEDS, a pruned centrality-based distributed algorithm, has the best performance.