This paper uses WITCH, an integrated assessment model with a game-theoretic structure, to explore the prospects for, and the stability of broad coalitions to achieve ambitious climate change mitigation action. Only coalitions including all large emitting regions are found to be technically able to meet a concentration stabilisation target below 550 ppm CO2eq by 2100. Once the free-riding incentives of non-participants are taken into account, only a "grand coalition" including virtually all regions can be successful. This grand coalition is profitable as a whole, implying that all countries can gain from participation provided appropriate transfers are made across them. However, neither the grand coalition nor smaller but still environmentally significant coalitions appear to be stable. This is because the collective welfare surplus from cooperation is not found to be large enough for transfers to offset the free-riding incentives of all countries simultaneously. Some factors omitted from the analysis, which might improve coalition stability, include the co-benefits from mitigation action, the costless removal of fossil fuel subsidies, as well as alternative assumptions regarding countries' bargaining behaviour.