Fish biodiversity sustains the resilience and productivity of fisheries, yet this biodiversity can be threatened by overharvest and depletion in mixed‐stock fisheries. Thus, the biodiversity that provides benefits may also make sustainable resource extraction more difficult, a key challenge in fisheries management. We simulated a mixed‐stock fishery to explore relationships between different dimensions of biodiversity and fishery performance relative to conservation and fishery objectives. Different dimensions of biodiversity (number of stocks, evenness, asynchrony among stocks, heterogeneity in stock productivity) exacerbated trade‐offs between fishery and conservation objectives. For example, fisheries targeting stock‐complexes with greater asynchrony, and to a lesser extent richness, had greater stability in harvest through time but also greater risks of overfishing weak stocks and reduced yield compared to less biodiverse stock‐complexes. These trade‐offs were ameliorated by increasing management control—the capacity of fishery managers to harvest specific stocks. To explore these trade‐offs in real‐world fisheries, we contrasted the fishing and population status of individual stocks within three major mixed‐stock sockeye salmon (Oncorhynchus nerka, Salmonidae) fisheries—Bristol Bay, Fraser River, and Skeena River. In general, the fisheries with lower management control had individual stocks that were more often being over‐ or under‐fished, compared with those with higher management control, though variation among regions in biodiversity, scale of management, and magnitude of habitat alteration likely also contribute to these relationships. Collectively, our findings emphasize that there is a need to extract less or regulate better in order to conserve and benefit from biodiversity in fisheries and other natural resource management systems.