Modelling experiments reducing surface temperatures via an idealized reduction of the solar constant have often been used as analogues for stratospheric aerosol injection (SAI), thereby implicitly assuming that solar dimming captures the essential physical mechanism through which SAI influences surface climate. While the omission of some important processes that otherwise operate under SAI was identified before, here we demonstrate that the imposed reduction in the incoming solar radiation also induces a different stratospheric dynamical response, manifested through a weakening of the polar vortex, that propagates from the stratosphere down to the troposphere. The coupled stratospheric-tropospheric response exerts a previously overlooked first-order influence on southern hemispheric surface climate in the solar dimming experiments, including on the position of the tropospheric jet and Hadley Circulation and thus, ultimately, precipitation patterns. This perturbation, opposite to that expected under SAI, highlights the need for caution when attributing responses in idealised experiments. In this dataset, we provide the model output used in the paper that demonstrate our findings.