Throughout human history, informal sanctions by peers were ubiquitous and played a key role in the enforcement of social norms and the provision of public goods. However, a considerable body of evidence suggests that informal peer sanctions cause large collateral damage and efficiency costs. This raises the question whether peer sanctioning systems exist that avoid these costs and whether other, more centralized, punishment systems are superior and will be preferred by the people. Here, we show that efficient peer sanctioning without much need for costly punishment emerges quickly if we introduce two relevant features of social life into the experiment: (i) subjects can migrate across groups with different sanctioning institutions and (ii) they have the chance to achieve consensus about normatively appropriate behavior. We also show that subjects universally reject peer sanctioning without a norm consensus opportunity - an institution that has hitherto dominated research in this field - in favor of our efficient peer sanctioning institution or an equally efficient institution where they delegate the power to sanction to an elected judge. Migration opportunities and normative consensus building are key to the quick emergence of an efficient culture of universal cooperation because the more prosocial subjects populate the two efficient institutions first, elect prosocial judges (if institutionally possible), and immediately establish a social norm of high cooperation. This norm appears to guide subjects' cooperation and punishment choices, including the virtually complete removal of antisocial punishment when judges make the sanctioning decision.