This paper develops a theory of the allocation of authority between two players who are in a “complex” partnership, that is, a partnership which produces impure public goods. We show that the optimal allocation depends on technological factors, the parties’ valuations of the goods produced, and the degree of impurity of these goods. When the degree of impurity is large, control rights should be given to the main investor, irrespective of preference considerations. There are some situations in which this allocation is optimal even if the degree of impurity is very low as long as one party’s investment is more important than the other party’s. If the parties’ investments are of similar importance and the degree of impurity is large, shared authority is optimal with a greater share going to the low-valuation party. If the importance of the parties’ investments is similar but the degree of impurity is neither large nor small, the low-valuation party should receive sole authority. We analyze an extension in which side payments are infeasible. We check for robustness of our results in several dimensions, such as allowing for multiple parties or for joint authority, apply our results to interpret a number of complex partnerships, including those involving schools and child custody.