Consider a seller of a divisible good, facing several identical buyers. The quality of the good may be low or high, and is the seller's private information. The seller has strictly convex preferences that satisfy a single-crossing property. Buyers compete by posting arbitrary menus of contracts. Competition is non-exclusive in that the seller can simultaneously and secretly trade with several buyers. We fully characterize conditions for the existence of an equilibrium. Equilibrium aggregate allocations are unique. Any traded contract must yield zero profit. If a quality is indeed traded, then it is traded efficiently. Depending on parameters, both qualities may be traded, or only one of them, or the market may break down completely to a no-trade equilibrium.