We introduce a distributed and tractable mechanism for the allocation of continuously divisible resources to agents, that locally maximises the (Nash) product of their individual welfare. The mechanism involves specific m-resources-at-a-time multilateral deals over bits of resources, termed infinitesimal Nash transfers. It provides an effective way of building "strong social alliances", where in a social alliance agents fully cooperate for the global interest of society, and a strong social alliance has near-optimal utilitarian and egalitarian social welfare, as understood in social choice and welfare economics. The mechanism is scalable, can be distributed amongst agents and can be used to support, e.g., fair trade.