We study unanimity bargaining among agents along a general river structure that is expressed by a geography matrix and who have access to limited local resources, cost functions that depend upon river inflow and own extraction, and quasi-linear preferences over water and money. Bargaining determines the water allocation and monetary transfers. We translate the legal principles of Absolute Territorial Sovereignty (ATS) and Unlimited Territorial Integrity (UTI) from International Water Law into our model. ATS and a strict interpretation of UTI result in disagreement outcomes that are feasible. And a second interpretation of UTI is translated into individual aspiration levels that are infeasible. For disagreement outcomes, we apply the asymmetric Nash bargaining solution. Common intuition that upstream and midstream countries always prefer the ATS principle to the strict UTI principle (while the opposite preference holds for the downstream country) is reversed for such countries with sufficient bargaining power. For individual aspiration levels, the agents have to compromise in order to agree and we apply the asymmetric Nash rationing solution. The participation constraints in the Nash rationing solution matter. In all cases the optimization problem is separable into two sub-problems: the efficient water allocation that maximizes utilitarian welfare; and the determination of monetary transfers.