Formation avoidance is one of the critical technologies for autonomous underwater vehicle (AUV) formations. To this end, a cooperative obstacle avoidance algorithm based on an improved artificial potential field method and a consistency protocol is proposed in this paper for the local obstacle avoidance problem of AUV formation. Firstly, for the disadvantage that the traditional artificial potential field method can easily fall into local minima, an auxiliary potential field perpendicular to the AUV moving direction is designed to solve the problem that AUVs can easily have zero combined force in the potential field and local minima. Secondly, the control law of AUV formation that keeps the speed and position consistent is designed for the problem that the formation will change during the local obstacle avoidance of the formation system. The control conflict problem of the combined algorithm of the artificial potential field law and the consistency protocol is solved by adjusting the desired formation of the consistency protocol through the potential field force. Finally, the bounded energy function demonstrates system convergence stability. The simulation verification confirmed that the AUV formation could achieve the convergence of the formation state under local obstacle avoidance.