Over the Southern Ocean (SO, 40°S–70°S), climate models have consistently underestimated solar reflection. Here we evaluate the relationship between cloud profiles, cloud phase and radiation over the SO in Coupled Model Intercomparison Project Phase 6 (CMIP6) models against Clouds and the Earth's Radiant Energy System and Cloud‐Aerosol Lidar and Infrared Pathfinder Satellite Observations. We find that the lack of solar reflection is slightly improved in CMIP6 models compared to CMIP5's, attributable to a better representation of cloud fraction and phase. We show that clouds have a different vertical structure and radiative effect south and north of where the 0°C isotherm meets the surface (∼55°S). Although the models capture the greater vertical extent of clouds south of 55°S, they fail to reproduce the observed increase in solar reflection, which we pinpoint to cloud phase biases. Increasing CMIP6 supercooled liquid cloud opacity should help reduce their persistent shortwave biases. Plain Language Summary Over the Southern Ocean, defined as the latitudinal band between 40° and 70°S, climate models have consistently overestimated the amount of absorbed solar radiation, mostly because of biases in the representation of clouds. Such biases in climate models are particularly problematic because they may affect the radiative response of clouds to climate warming. We find that the newest generation of climate models better represents clouds than the previous one, compared to satellite observations. We show that clouds have a different vertical structure and radiative properties south and north of the 0°C surface isotherm, around 55°S. Although the models capture the greater vertical extent of clouds south of 55°S, they fail to reproduce the observed increase in solar reflection by clouds there. Finally, we report that increasing the opacity of liquid clouds at subzero temperatures (between 0° and −40°C) should help reduce persistent solar radiation biases attributable to clouds, south of 55°S, in the newest generation of climate models. Key Points The lack of shortwave reflection is somewhat corrected in Coupled Model Intercomparison Project Phase 6 (CMIP6) models, attributable to a better representation of cloud fraction and cloud phase The cloud fraction and radiative effect behavior is different north and south of 55°S, which is where the 0°C isotherm meets the surface Contrary to observations, CMIP6 clouds are less reflective south of 55°S, where boundary layer clouds are often topped by mid‐level clouds