It has been suggested that increased stratospheric sulfate aerosol loadings following large, low latitude volcanic eruptions can lead to wintertime warming over Eurasia through dynamical stratosphere-troposphere coupling. We here investigate the proposed connection in the context of hypothetical future stratospheric sulfate geoengineering in the Geoengineering Large Ensemble simulations. In those geoengineering simulations, we find that stratospheric circulation anomalies that resemble the positive phase of the Northern Annular Mode in winter are a distinguishing climate response which is absent when increasing greenhouse gases alone are prescribed. This stratospheric dynamical response projects onto the positive phase of the North Atlantic Oscillation, leading to associated side effects of this climate intervention strategy, such as continental Eurasian warming and precipitation changes. Seasonality is a key signature of the dynamically driven surface response. We find an opposite response of the North Atlantic Oscillation in summer, when no dynamical role of the stratosphere is expected. The robustness of the wintertime forced response stands in contrast to previously proposed volcanic responses.