Time synchronization is important for a variety of applications in wireless networks including scheduling communication resources, interference avoidance, and data fusion. This paper analyzes the problem of synchronizing nodes in a time-division-duplexed wireless network via consensus methods using only acknowledged message exchanges over existing network traffic. The nodes are assumed to communicate synchronization information randomly and asymmetrically, reflecting the random nature of the timing and the source of synchronization information gleaned from or embedded in existing network traffic. The synchronization method accounts for non-negligible propagation delays which can be disruptive to consensus techniques. To characterize both transient and asymptotic consensus behavior, general results are presented providing necessary and sufficient conditions for monotonic mean squared convergence of a distance from consensus metric at an exponential rate. While the general results apply to a broad class of random consensus models, two models are analyzed in detail: i) random asymmetric gossip and ii) fully connected random broadcast. Bounds are derived for the steady-state distance from consensus in the presence of estimation error. Numerical results are also presented verifying the analysis under different network topologies.