Based on the mean-variance criterion, this paper studies the robust optimal reinsurance problem under the competition between an insurance company and a reinsurance company. The insurance company operates $ n $ kinds of dependent insurance businesses, and it buys reinsurance for each insurance business to reduce the claim risk. Through relative performance, this paper quantifies the competition between the insurance company and the reinsurance company. The insurance company's goal is to choose an optimal reinsurance strategy to minimize the risk when the mean of terminal wealth is given in the worst market situation. By using the theory of stochastic control and stochastic dynamic programming, this paper establishes the Hamilton-Jacob-Bellman-Isaacs (HJBI) equation. Furthermore, by solving HJBI equation and using Lagrange duality theory, this paper obtains the explicit solution for the robust optimal reinsurance strategy. Finally, the influence of model parameters on the robust optimal reinsurance strategy and efficient frontier is explained by numerical experiments. The research results can guide insurance companies to adopt optimal reinsurance strategies to minimize the risks that they face when operating a variety of insurance businesses.