Climate change strains public finances, eroding sovereigns’ capacity to fund climate policies. We build a forward-looking stress-test framework for sovereign debt that links an integrated climate-economy assessment model to stochastic debt-sustainability analysis using IPCC narrative scenarios. The framework pairs shared socioeconomic pathways and warming narratives with low- and high-climate-impact functions to calibrate aleatory scenario trees for the economic, fiscal, and financial determinants of debt. It projects debt dynamics under optimal debt financing. Stress-testing eight economies across climate vulnerability and fiscal space, we find adverse shifts of the cost-risk frontier of debt financing, with material increases in expected costs and high regional heterogeneity. Debt trajectories trend upward from mid-century, and stabilizing them requires fiscal consolidation that exceeds a feasibility threshold for the more vulnerable economies. Adaptation moderates the impact and breaks even when governments finance about one-third of the cost, with the private sector covering the rest, yet it cannot alone restore sustainability. Maintaining current public spending in the face of climate-related damages appears impossible. Thus, climate change emerges as a first-order risk for fiscal authorities.