We study the robust communication complexity of maximum matching. Edges of an arbitrary $n$-vertex graph $G$ are randomly partitioned between Alice and Bob independently and uniformly. Alice has to send a single message to Bob such that Bob can find an (approximate) maximum matching of the whole graph $G$. We specifically study the best approximation ratio achievable via protocols where Alice communicates only $\widetilde{O}(n)$ bits to Bob. There has been a growing interest on the robust communication model due to its connections to the random-order streaming model. An algorithm of Assadi and Behnezhad [ICALP'21] implies a $(2/3+ε_0 \sim .667)$-approximation for a small constant $0 < ε_0 < 10^{-18}$, which remains the best-known approximation for general graphs. For bipartite graphs, Assadi and Behnezhad [Random'21] improved the approximation to .716 albeit with a computationally inefficient (i.e., exponential time) protocol. In this paper, we study a natural and efficient protocol implied by a random-order streaming algorithm of Bernstein [ICALP'20] which is based on edge-degree constrained subgraphs (EDCS) [Bernstein and Stein; ICALP'15]. The result of Bernstein immediately implies that this protocol achieves an (almost) $(2/3 \sim .666)$-approximation in the robust communication model. We present a new analysis, proving that it achieves a much better (almost) $(5/6 \sim .833)$-approximation. This significantly improves previous approximations both for general and bipartite graphs. We also prove that our analysis of Bernstein's protocol is tight.