When different flooding drivers co-occur, they can cause compound floods. Despite the potential impact of compound flooding, few studies have projected how the joint probability of flooding drivers may change. Furthermore, existing projections may not be very robust, as they are based on only 5 to 6 climate model simulations. Here, we use a large ensemble of simulations from the Coupled Model Intercomparison Project 6 (CMIP6) to project changes in the joint probability of extreme storm surges and precipitation at European tide gauges under a medium and high emissions scenario, enabled by data-proximate cloud computing and statistical storm surge modeling. We find that the joint probability will increase in the northwest and decrease in most of the southwest of Europe. Averaged over Europe, the absolute magnitude of these changes is 36% to 49% by 2080, depending on the scenario. The large-scale changes in the joint probability of extreme wind speed and precipitation are similar, but locally, differences between the changes in the two types of compound extremes can exceed the changes themselves. Due to internal climate variability and inter-model differences, projections based on only 5 to 6 random climate model simulations have a probability of higher than 10% to differ qualitatively from projections based on all CMIP6 simulations in multiple regions, especially under the medium emissions scenario and earlier in the 21st century. Therefore, our results provide a more robust and less uncertain representation of changes in the potential for compound flooding in Europe than previous projections.