Extreme events can severely impact marine organisms and ecosystems. Of particular concern are multivariate compound events, namely when conditions are simultaneously extreme for multiple ocean ecosystem stressors. In 2013-2015 for example, an extensive marine heatwave (MHW), known as the Blob, co-occurred locally with extremely low net primary productivity (NPPX) and negatively impacted marine life in the northeast Pacific. Yet, little is known about the characteristics and drivers of such multivariate compound MHW-NPPX events. Using five different satellite-derived net primary productivity (NPP) estimates and large-ensemble-simulation output of two widely used and comprehensive Earth system models, the Geophysical Fluid Dynamics Laboratory (GFDL) ESM2M-LE and Community Earth System Model version 2 (CESM2-LE), we assess the present-day distribution of compound MHW-NPPX events and investigate their potential drivers on the global scale. The satellite-based estimates and both models reveal hotspots of frequent compound events in the center of the equatorial Pacific and in the subtropical Indian Ocean, where their occurrence is at least 3 times higher (more than 10 d yr-1) than if MHWs (temperature above the seasonally varying 90th-percentile threshold) and NPPX events (NPP below the seasonally varying 10th-percentile threshold) were to occur independently. However, the models show disparities in the northern high latitudes, where compound events are rare in the satellite-based estimates and GFDL ESM2M-LE (less than 3 d yr-1) but relatively frequent in CESM2-LE. In the Southern Ocean south of 60 S, low agreement between the observation-based estimates makes it difficult to determine which of the two models better simulates MHW-NPPX events. The frequency patterns can be explained by the drivers of compound events, which vary among the two models and phytoplankton types. In the low latitudes, MHWs are associated with enhanced nutrient limitation on phytoplankton growth, which results in frequent compound MHW-NPPX events in both models. In the high latitudes, NPPX events in GFDL ESM2M-LE are driven by enhanced light limitation, which rarely co-occurs with MHWs, resulting in rare compound events. In contrast, in CESM2-LE, NPPX events in the high latitudes are driven by reduced nutrient supply that often co-occurs with MHWs, moderates phytoplankton growth, and causes biomass to decrease. Compound MHW-NPPX events are associated with a relative shift towards larger phytoplankton in most regions, except in the eastern equatorial Pacific in both models, as well as in the northern high latitudes and between 35 and 50 S in CESM2-LE, where the models suggest a shift towards smaller phytoplankton, with potential repercussions on marine ecosystems. Overall, our analysis reveals that the likelihood of compound MHW-NPPX events is contingent on model representation of the factors limiting phytoplankton production. This identifies an important need for improved process understanding in Earth system models used for predicting and projecting compound MHW-NPPX events and their impacts.