Although qualified-majority voting is possible, member states in the Council of the European Union (EU) still adopt most policies by consensus. The agent-based model of coalition building in multilateral negotiations presented here addresses this puzzle. The model demonstrates that consensual decisions may emerge as an unintended by-product of government representatives' desire to form blocking coalitions. A qualitative case study demonstrates the plausibility of the model's assumptions and resulting coalition-building dynamics. Moreover, a quantitative test shows that the model's predictions correspond closely to the observed consensus rates. Finally, computational experiments predict a positive effect of the voting threshold but no effect of increases in membership on winning coalition size, which has important practical implications for institutional design and enlargement policy.