Co-production has emerged as one of the key concepts in understanding knowledge-policy interactions and is associated with involvement, for example, of users of public services in their design and delivery. At a time of permacrisis, the need for transformative evidence-based policymaking is urgent and great. This is particularly important in highly distressed ‘left-behind’ communities targeted by the UK Government for Levelling Up, which constitutes an attempt to improve the infrastructural, economic, social and health of less affluent parts of the UK. Often, policymakers regard the transformative policies capable of addressing these crises as beyond the ‘Overton Window’, which describes a range of policies in the political centre that are acceptable to the public (Lehman 2010). This window of opportunity can shift to encompass different policies, but movement is slow and policymakers generally believe that significant change lies outside. In this article, we build on recent debates in Evidence & Policy on co-production by outlining an embryonic approach to overcoming this Overton Window-based roadblock in evidence-based policymaking: adversarial co-production, which involves working with opponents of evidence-based policy to develop means of persuading potential beneficiaries to support introduction. This emerging approach has been deployed in examination of public preferences with regard to welfare reform, but can be applied to a wide range of policy areas. We outline briefly the history of co-production, before setting out the process by which adversarial co-production was developed. We then describe the impact of adversarial co-production on public preferences on basic income (BI). This enables us to set out challenges and opportunities for those with an interest in addressing our crises, serving to stimulate genuine debate on longstanding assumptions about the limits of evidence-based policy and public opinion.