We propose a model in which money performs an essential role in the process of exchange, despite the presence of a multilateral clearing house. Agents are assumed to be anonymous and unable to make binding commitments. The clearing house can detect deviations but it cannot identify the individual deviator, hence, it punishes all traders collectively. The records of past deviations can be kept for a limited amount of time, after which they are wiped out. These features are enough to make room for a record-keeping device, such as money, that strictly improves the functioning of the clearing house.