A product-lubricated multiphase pump for pumping a multiphase mixture containing hydrocarbon includes a separation system and a supply system. To achieve a good cooling and lubrication of pump elements without a lot of operating media and operating medium circuits, the separation system has a first separation stage for at least partly separating at least a portion of the multiphase mixture into a plurality of phase-enriched components and the supply system supplies a liquid-enriched liquid component as lubricant to a pump element to be lubricated.