The flow in a source-fed f-plane basin drained through a strait is explored using a single-layer (reduced gravity) shallow-water numerical model that resolves the hydraulic flow within the strait. The steady upstream basin circulation is found to be sensitive to the nature of the mass source (uniform downwelling, localized downwelling, or boundary inflow). In contrast, the hydraulically controlled flow in the strait is nearly independent of the basin circulation and agrees very well with the Gill-theory solution obtained using the strait geometry and the numerically determined average potential vorticity in the strait entrance region. This Gill solution, however, gives a unique value of the upstream boundary layer flux splitting that does not agree with any of the full numerical solutions. The coupled basin–strait system is shown to select an average overflow potential vorticity corresponding to the Gill solution with maximum fluid depth on the strait boundaries. This state also corresponds to one of maximal upstream basin potential energy. This result is robust to changes in the basin geometry, strait characteristics, the dissipation parameter (linear drag), and the net mass flux. The nonunique relation between basin conditions and overflow transport is significant with regard to deep overflow transport monitoring. It is shown that the potential vorticity selection leads to overflow, or “weir,” transport relations that are well approximated by the zero potential vorticity theory. However, accurate estimates of the transport can only be obtained if conditions within the strait entrance region, and not the basin, are used.