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Tropospheric and stratospheric tropical temperature trends in recent decades have been notoriously hard to simulate using climate models, notably in the upper troposphere. Aside from the warming trend itself, this has broader implications, e.g. atmospheric circulation trends depend on latitudinal temperature gradients. In this study, tropical temperature trends in the CMIP6 models are examined, from 1979 to 2014, and contrasted with trends from the RICH/RAOBCORE radiosondes, and the ERA5/5.1 reanalysis. As in earlier studies, we find considerable warming biases in the CMIP6 modeled trends, and we show that these biases are linked to biases in surface temperature (these models simulate an unrealistically large global warming). We also uncover previously undocumented biases in the lower-middle stratosphere: the CMIP6 models appear unable to capture the time evolution of stratospheric cooling, which is non-monotonic owing to the Montreal Protocol. Finally, using models with large ensembles, we show that their standard deviation in tropospheric temperature trends, which is due to internal variability alone, explains $\sim$50\% ($\pm$20\%) of that from the CMIP6 models.