We model problems of allocating disputed properties as generalized exchange economies. Therein, agents have preferences and claims over multiple goods, and the social endowment of each good may not be sufficient to satisfy all individual claims. We focus on market-based allocation rules that impose a two-step procedure: assignment of rights based on claims first, and voluntary exchange based on the assigned rights afterwards. We characterize three focal egalitarian rights-assignment rules that guarantee that the allocation rules are fair. We apply our results to problems of greenhouse gas emissions and contested water rights.