Policy processes are affected by how policymakers assess public support for a policy. But is public support for a given policy itself affected by characteristics of the policy process, such as cooperation or confrontation amongst policy actors? Specifically, if different branches of government hold conflicting positions on a given policy, do clashes affect public support for the policy? To address this question, we exploit an unexpected clash amongst the executive and judiciary in New Delhi, between survey waves, over exemptions for women in the context of the odd–even rule, a policy intervention to reduce air pollution from transportation. We find that public support for the contested policy was not undermined by the executive–judiciary clash. However, the clash polarised public opinion by gender, based upon the policy exemptions. Our findings shed new light on the broader question of how conflicts amongst different parts of government influence mass public policy preferences.