The magnetohydrodynamic flow of a liquid metal through a manifold that feeds an array of electrically coupled rectangular ducts with thin conducting walls is investigated. This geometry is typical of an inlet/outlet manifold servicing arrays of poloidal coolant channels in tokamak self-cooled blankets. The interaction parameter and Hartmann number are assumed to be large, whereas the magnetic Reynolds number is assumed to be small. Under these assumptions, which are relevant to liquid-metal flows in self-cooled tokamak blankets, viscous and inertial effects are confined to very thin boundary layers adjacent to the walls. The analysis for obtaining three-dimensional solutions outside these layers is described, and numerical solutions are presented. Electrical coupling between the common manifold and the coolant ducts, as well as coupling among the coolant ducts themselves, necessitates simultaneous solutions for the multiple channels, and uniquely determines the partition of the total flow rate among the coolant ducts. Control of flow partition that may be required for optimal cooling of the first wall and blanket is demonstrated and discussed. The pressure drop resulting from the disturbance associated with the manifold is calculated and is shown to be minimal.