This paper presents a real-time kinematically synchronous planning method for the collaborative manipulation of a multi-arms robot with physical coupling based on the self-organizing competitive neural network. This method defines the sub-bases for the configuration of multi-arms to obtain the Jacobian matrix of common degrees of freedom so that the sub-base motion converges along the direction for the total pose error of the end-effectors (EEs). Such a consideration ensures the uniformity of the EE motion before the error converges completely and contributes to the collaborative manipulation of multi-arms. An unsupervised competitive neural network model is raised to adaptively increase the convergence ratio of multi-arms via the online learning of the rules of the inner star. Then, combining with the defined sub-bases, the synchronous planning method is established to achieve the synchronous movement of multi-arms robot rapidly for collaborative manipulation. Theory analysis proves the stability of the multi-arms system via the Lyapunov theory. Various simulations and experiments demonstrate that the proposed kinematically synchronous planning method is feasible and applicable to different symmetric and asymmetric cooperative manipulation tasks for a multi-arms system.