Design of reliable wireless backhaul networks is challenging due to the inherent vulnerability of wireless backhauling to random fluctuations of the wireless channel. Considerable studies deal with modifying and designing the network topology to meet the reliability requirements in a cost-efficient manner. However, these studies ignore the correlation among link failures, particularly those caused by weather disturbances. Consequently, the resulting topology designs may fail to meet the network reliability requirements under correlated failure scenarios. To fill this gap, we study the design of cost-efficient and reliable wireless backhaul networks under correlated failures with a focus on rain disturbances. We first propose a new model to consider the pairwise correlation amongf links along a path. The model is verified on real data, indicating an approximation closer to reality than the existing independent failure model. Second, we model the correlation among different paths by defining a penalty cost. Considering the newly formalized link and path correlation, we formulate the correlation-aware network topology design problem as a quadratic integer program to find the optimal solutions. Two lightweight heuristic algorithms are developed to find near-optimal solutions within reasonable time. Performance evaluation shows that correlation-aware design substantially improves the resiliency under rain disturbances at a slightly increased cost compared to independent failure approaches.