One popular and efficient scheme for solving combinatorial optimization problems over graphical models exactly is depth-first Branch and Bound. However, when the algorithm exploits problem decomposition using AND/OR search spaces, its anytime behavior breaks down. This article 1) analyzes and demonstrates this inherent conflict between effective exploitation of problem decomposition (through AND/OR search spaces) and the anytime behavior of depth- first search (DFS), 2) presents a new search scheme to address this issue while maintaining desirable DFS memory properties, and 3) analyzes and demonstrates its effectiveness through comprehensive empirical evaluation. Our work is applicable to any problem that can be cast as search over an AND/OR search space.