A Coalescing-Branching Random Walk (CoBra) is a natural extension to the standard random walk on a graph. The process starts with one pebble at an arbitrary node. In each round of the process every pebble splits into k pebbles, which are sent to k random neighbors. At the end of the round all pebbles at the same node coalesce into a single pebble. The process is also similar to randomized rumor spreading, with each informed node pushing the rumor to k random neighbors each time it receives a copy of the rumor. Besides its mathematical interest, this process is relevant as an information dissemination primitive and a basic model for the spread of epidemics. We study the cover time of CoBra walks, which is the time until each node has seen at least one pebble. Our main result is a bound of $O( logn/φ$) rounds with high probability on the cover time of a CoBra walk with $k = 2$ on any regular graph with n nodes and conductance $φ$. This bound improves upon all previous bounds in terms of graph expansion parameters (Dutta et al. [13], Mitzenmacher et al. [27], Cooper et al. [8, 9]). Moreover, we show that for any connected regular graph the cover time is $O (n log n)$ with high probability, independently of the expansion. Both bounds are asymptotically tight. Since our bounds coincide with the worst-case time bounds for Push rumor spreading on regular graphs until all nodes are informed, this raises the question whether CoBra walks and Push rumor spreading perform similarly in general. We answer this negatively by separating the cover time of CoBra walks and the rumor spreading time of Push by a super-polylogarithmic factor on a family of tree-like regular graphs.