We find that stratospheric warming suffices to reproduce a poleward jet shift. However, the hypothesized mechanism (a strengthening of the stratospheric winds, in balance with the warming anomaly, which in turn stimulates an annular mode response in the troposphere) does not hold quantitatively. Specifically, we find that the warm anomaly is spread into the subtropics by a Hadley Cell-like response of the zonally symmetric circulation, and then to higher latitudes by changes in wave breaking. Thus, the final response of the stratosphere fundamentally depends on eddy interactions. The tropospheric jet shift also depends critically on changes in wave breaking, evolving in concert with the extratropical stratosphere. The coupling bears some similarity to that of the annular mode—i.e., a stronger polar vortex is associated with a poleward jet shift in the troposphere—but the annular mode cannot explain the seasonal variation of the response, or its sensitivity to the model climatology. Lastly, we discuss implications for comprehensive models.