In the framework of the second phase of the Coordinated Ocean-ice Reference Experiments (CORE-II), we present an analysis of the representation of the Antarctic Circumpolar Current (ACC) and Southern Ocean meridional overturning circulation (MOC) in a suite of seventeen global ocean–sea ice models. We focus on the mean, variability and trends of both the ACC and MOC over the 1958–2007 period, and discuss their relationship with the surface forcing. We aim to quantify the degree of eddy saturation and eddy compensation in the models participating in CORE-II, and compare our results with available observations, previous fine-resolution numerical studies and theoretical constraints. Most models show weak ACC transport sensitivity to changes in forcing during the past five decades, and they can be considered to be in an eddy saturated regime. Larger contrasts arise when considering MOC trends, with a majority of models exhibiting significant strengthening of the MOC during the late 20th and early 21st century. Only a few models show a relatively small sensitivity to forcing changes, responding with an intensified eddy-induced circulation that provides some degree of eddy compensation, while still showing considerable decadal trends. Both ACC and MOC interannual variabilities are largely controlled by the Southern Annular Mode (SAM). Based on these results, models are clustered into two groups. Models with constant or two-dimensional (horizontal) specification of the eddy-induced advection coefficient κ show larger ocean interior decadal trends, larger ACC transport decadal trends and no eddy compensation in the MOC. Eddy-permitting models or models with a three-dimensional time varying κ show smaller changes in isopycnal slopes and associated ACC trends, and partial eddy compensation. As previously argued, a constant in time or space κ is responsible for a poor representation of mesoscale eddy effects and cannot properly simulate the sensitivity of the ACC and MOC to changing surface forcing. Evidence is given for a larger sensitivity of the MOC as compared to the ACC transport, even when approaching eddy saturation. Future process studies designed for disentangling the role of momentum and buoyancy forcing in driving the ACC and MOC are proposed.