Multi-issue negotiations are a central component of many important coordination challenges. Almost all previous work in this area has assumed that negotiation issues are independent, making it relatively easy to find highquality agreements. In many real-world problem domains, however, issues are interdependent, making hard to find good agreement due to the nonlinearity of the agent?s utility functions. The key challenge, in this context, is finding high-quality agreements without making unrealistic demands concerning how much agents reveal about their utilities. In this paper, we propose a protocol wherein the negotiating agents, working with the mediator, progress through a multi-phase narrowing of the space of possible agreements. We show that our method outperforms existing methods in large nonlinear utility spaces.