The circulation of the subtropical gyres can be decomposed into a horizontal recirculation along contours of constant Bernoulli potential and an overturning circulation across these contours. While the geometry and topology of Bernoulli contours is more complicated in the subtropical gyres than in the Southern Ocean, these subtropical overturning circulations are very much analogous to the overturning cell found in the Southern Ocean. This analogy is formalised through an exact integral constraint, including the rectified effects of transient eddies. The constraint can be interpreted either in terms of vertical fluxes of potential vorticity, or equivalently as an integral buoyancy budget for an imaginary fluid parcel recirculating around a closed Bernoulli contour. Under conditions of vanishing buoyancy and mechanical forcing, the constraint reduces to a generalised non-acceleration condition, under which the Eulerian-mean and eddy-induced overturning circulations exactly compensate. The terms in the integral constraint are diagnosed in an eddy-permitting ocean model in both the North Pacific subtropical gyre and the Southern Ocean. The extent to which the Eulerian-mean and eddy-induced overturning circulations compensate is discussed in each case.