Meeting climate goals requires radical changes in the consumption behaviour of individuals. This necessitates an understanding of how the diffusion of low-carbon behaviour will occur. The speed and interdependency of these changes in behavioural choices may be modulated by individuals’ culture. We develop an agent-based model to study how behavioural decarbonisation interacts with longer-term cultural change, composed of individuals with multiple behaviours that evolve due to imperfect social learning in a social network. Using the definition of culture as socially transmitted information, we represent individuals’ environmental identity as an aggregation of attitudes towards multiple relevant behaviours. The strength of interaction between individuals is determined by the similarity in their environmental identity, leading to inter-behavioural dependency and spillovers in green attitudes. Our results show that the initial distribution of agent attitudes towards behaviours and asymmetries in social learning, such as confirmation bias, are the main drivers of model dynamics, helping to generate awareness of what roadblocks may appear to deep decarbonisation. To assess the impact of culture beyond a purely diffusive regime, we introduce green influencers as a minority of individuals who broadcast a green attitude. The greatest emissions reduction is achieved with the inclusion of culture, relative to a behavioural independence case, and with low confirmation bias. However, green influencers fail to achieve deep behavioural decarbonisation through solely voluntary action. We identify areas for further research regarding how culture, through inter-behavioural dependence, may be leveraged for climate policy.