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Underwater Acoustic Sensor Networks (UASNs) have become one of the promising technologies for exploring underwater natural resources and collecting scientific data from the aquatic environment. As obstacles hinder the communications among sensor nodes in UASNs, designing an effective bypass routing protocol to avoid obstacles is an urgent need. Moreover, the sensor nodes are typically powered by batteries, which are difficult to replace, restricting the network lifetime of UASNs. In this paper, an Energy-efficient and Obstacle-Avoiding Routing protocol (EOAR) is proposed not only to address the issue of marine animals acting as obstacles that interfere with communications, but also to balance the network energy according to the residual energy. In the EOAR protocol, when the current node perceives the existence of marine animals, the interference area of the animal-nodes is first calculated using the underwater acoustic channel model, and then the candidate forwarding relay set of the current node is obtained according to the constraint conditions. The optimal candidate forwarding relay is determined by a fuzzy logic-based forwarding relay selection scheme based on considering the three parameters of the candidate forwarding relay, which includes the propagation delay, the included angle between two neighbor nodes, and the residual energy. Furthermore, in order to solve the problem of energy waste caused by packet collision, we use a priority-based forwarding method to schedule the packet transmission from the candidate forwarding relay to the destination node. The proposed EOAR protocol is simulated on the Aqua-sim platform and the simulation results show that proposed protocol can increase the packet delivery ratio by 28.4% and 11.8% and can reduce the energy consumption by 53.4% and 32.7% and, respectively, comparing with the hop-by-hop vector-based forwarding routing protocol (HHVBF) and void handling using geo-opportunistic routing protocol (VHGOR).