Two-way coupling between the stratosphere and troposphere is an important source of subseasonal  predictability and can open windows of opportunity for improved forecasts. However, operational models used for subseasonal forecasting suffer from systematic biases due to insufficient resolution or poorly performing parameterizations: they underestimate the strength of upward  coupling from the troposphere to  the stratosphere, they overestimate downward coupling of the lower stratosphere to near-surface polar cap height, and the polar lower stratosphere is too cold. Using data from 22 operational subseasonal models, we explore the implications of these biases on the skill of the Northern and Southern Annular Modes (NAM and SAM) in both the troposphere and stratosphere and for regional surface impacts. Updated or upgraded model versions from nearly all modeling centers show higher skill in both the stratospheric and tropospheric NAM and SAM as compared to their earlier versions, and models with relatively higher skill for the stratospheric annular modes generally have higher skill for the tropospheric annular modes as well. Near-surface AM skill is significantly higher for initializations during strong vortex states, and to a lesser degree during weak vortex states. Models with particularly poor upward coupling strengths and with too-weak climatological heat flux at 100 hPa suffer from reduced stratospheric NAM and SAM skill. In contrast, mean state biases in the stratospheric polar vortex are relatively unimportant for stratospheric NAM and SAM skill. Even as downward coupling over the North Atlantic and Antarctica tends to be handled well by these models, they produce an overly zonal response with a  fictitious center of action in the midlatitude North Pacific, while underestimating the midlatitude near-surface temperature impacts over Eurasia, Australia, and South America. Downward coupling  in the North Pacific is particularly pronounced in models with overly strong variance in this region, and models underestimate the wave-3 pattern in Southern Hemisphere (SH) midlatitudes by a  factor of three. These biases in downward coupling mirror biases in the midlatitude structure of the tropospheric  NAM and SAM. Overall, models with finer vertical resolution near the tropopause, with a better representation of tropospheric quasi-stationary waves, and with a better representation of these coupling processes, are more skillful in both the stratosphere and troposphere. These findings have implications for future research directions of the Stratospheric Network for the Assessment of Predictability.