Coupled climate models predict density-driven weakening of the Atlantic meridional overturning circulation (AMOC) under greenhouse gas forcing, with considerable spread in the response between models. There is also a large spread in the predicted increase of the southern annular mode (SAM) index across these models. Regression analysis across model space using 11 non-eddy-resolving models suggests that up to 35% of the intermodel spread in the AMOC response may be associated with uncertainty in the magnitude of the increase in the SAM. Models with a large, positive SAM index response generally display a smaller weakening of the AMOC under greenhouse gas forcing. The initial AMOC strength is also a major cause of intermodel spread in its response to climate change. The increase in the SAM acts to reduce the weakening of the AMOC over the next century by around ⅓, through increases in wind stress over the Southern Ocean, northward Ekman transport, and upwelling around Antarctica. The SAM response is also related to an increase in the northward salt flux across 30°S and to salinity anomalies in the high-latitude North Atlantic. These provide a positive feedback by further reinforcement of the AMOC. The results suggest that, compared with the real ocean where eddies oppose wind-driven changes in Southern Ocean circulation, climate models underestimate the effects of anthropogenic climate change on the AMOC.