Nowadays, wireless multimedia sensor networks (WMSNs) are used in various applications. An energy-efficient and robust routing protocol is essential for WMSNs because the quality of service is important for traffic-intensive multimedia data, such as images and videos. A WMSN with multiple sinks allows cluster heads (CHs) to deliver the collected data to the nearest sink, thereby mitigating the delivery overhead. In this study, we propose a novel evolutionary-game-based routing (EGR) protocol for WMSNs with multiple sinks, in which the evolutionary game theory is exploited for selecting CHs. In EGR, an algorithm to mitigate data redundancy, based on the overlapping field of views of the multimedia sensor nodes, is also presented. This algorithm decreases the number of redundant transmissions, thereby increasing energy efficiency and network performance. According to the performance evaluation results of this study, the proposed EGR significantly outperforms the state-of-art protocols in terms of energy efficiency, end-to-end delay, packet delivery ratio, cluster formation time, and network lifetime.