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Large-scale cooperation in one-time interactions among unrelated individuals is a widespread phenomenon in human societies and has drawn much scholarly attention. Reputational concerns can support such cooperation, but the informational content of reputations varies across cultures. In a punitive reputation mechanism, individuals punish defectors in order to gain a reputation as being “tough”, securing that future co-players will cooperate. In cultures with image scoring, defectors gain a negative image, triggering other group members to avoid interactions. Here, we study what happens when closed systems, characterized by different reputational mechanisms, interact. Using a game theoretic model and computer simulations, we show that injecting pools of individuals from a “culture of honor”, that achieves cooperation through punishment, into a culture of image scoring, can lead to widespread intergroup conflict. Cooperation is a much more likely outcome if the cultures that interact use a similar reputational mechanism. The theoretical framework helps us explain conflicts between natives and certain immigrant groups. Classifying host and source countries of immigrants according to their reputational mechanism, we show that our model helps to predict immigrant crime rates as well as exclusionary attitudes of natives, even after accounting for country-level crime rates and attitudes.