We stress-test sovereign debt for representative countries under the IPCC marker narrative scenarios of climate change using a stochastic debt sustainability analysis that integrates a coupled climate-economy model with debt financing scenario optimization. Our approach combines socioeconomic and climate pathways with calibrated aleatory scenario trees of economic, fiscal, and financial variables, generating forward-looking debt projections over the century. These projections incorporate climate-induced damages to economic growth spanning the broad spectrum of impact functions from the literature. Our findings reveal significant risks to sovereign debt stability, particularly under high-impact climate damage scenarios. We assess whether adaptation investments or fiscal consolidation can mitigate potential climate-debt crises. Notably, public financing of reactive adaptation can enhance debt sustainability, making it a justified expenditure. However, maintaining public spending at current levels while ensuring debt sustainability appears infeasible under climate impacts.