The Arctic winter atmospheric boundary layer often features strong and persistent low-level stability which arises from longwave radiative cooling of the surface during the polar night. This stable stratification results in a positive lapse rate feedback, which is a major contributor to Arctic amplification. A second state, with cloudy conditions, weaker stability, and near-zero net surface longwave flux is also observed. Previous work has shown that many CMIP5 models fail to appropriately partition water between liquid and ice phases in mixed-phase clouds, leading to a lack of this cloudy state. In this study, we assess the representation of the Arctic winter atmospheric boundary layer over sea ice in global climate models contributing to the latest phase of the Coupled Model Intercomparison Project (CMIP6). We compare boundary-layer process relationships in these models to those in surface-based and radiosonde observations collected during the MOSAiC (2019-2020) and SHEBA (1997-1998) expeditions, and by North Pole drifting stations (1955-1991). The majority of CMIP6 models fail to realistically represent the cloudy state over winter Arctic sea ice. Despite this, CMIP6 multi-model mean low-level stability falls within the observational range, and models mostly capture the observed dependence of low-level stability on near-surface air temperature and wind speed. CMIP6 models predict winter low-level stability to decline with Arctic warming, and to fall below zero by 2100 under the SSP2-4.5 scenario. Our results highlight the lack of the cloudy state as an important limitation on representing a realistic Arctic winter boundary layer in many CMIP6 models.