We consider an exchange economy in which a seller can trade an endowment of a divisible good whose quality she privately knows. Buyers compete in menus of non-exclusive contracts, so that the seller may choose to trade with several buyers. In this context, we show that an equilibrium always exists and that aggregate equilibrium allocations are generically unique. Although the good offered by the seller is divisible, aggregate equilibrium allocations exhibit no fractional trades. In equilibrium, goods of relatively low quality are traded at the same price, while goods of higher quality may end up not being traded at all if the adverse selection problem is severe. This provides a novel strategic foundation for Akerlof’s (1970) results, which contrasts with standard competitive screening models postulating enforceability of exclusive contracts. Latent contracts that are issued but not traded in equilibrium turn out to be an essential feature of our construction.