Global storm-resolving models (GSRMs) are the next avenue of climate modelling. Among them is the 5-km Icosahedral Nonhydrostatic Weather and Climate Model (ICON). The high resolution allows for parameterizations of convection and clouds to be avoided. Standard-resolution models have substantial cloud biases over the Southern Ocean (SO), affecting radiation and sea surface temperature. We evaluated SO clouds in ICON and the ERA5 and MERRA-2 reanalyses. The SO is dominated by low clouds, which cannot be observed accurately from space due to overlapping clouds, attenuation, and ground clutter. Instead, we analysed about 2400 days of lidar observations from 31 voyages and a station using a ground-based lidar simulator. ICON and the reanalyses underestimate the total cloud fraction by about 10 and 20%, respectively. ICON and ERA5 overestimate the cloud occurrence peak at about 500 m, potentially explained by their lifting condensation levels being too high. The reanalyses strongly underestimate near-surface clouds or fog. MERRA-2 tends to underestimate cloud occurrence at all heights. Less stable conditions are the most problematic for ICON and the reanalyses. In daily cloud cover, ICON and the reanalyses tend to be about 1 and 2 oktas clearer, respectively. Compared to radiosondes, potential temperature is accurate in the reanalyses, but ICON underestimates stability over the low-latitude SO and too humid in the boundary layer. MERRA-2 is too humid at all heights. SO cloud biases remain a substantial issue in the GSRM, but are an improvement over the lower-resolution reanalyses. Explicitly resolved convection and cloud processes were not enough to address the model cloud biases.