Climate change strains public finances through direct local economic effects, yet interconnected economies also face indirect climate exposure via global value chains. These spillovers can amplify fiscal costs but are widely ignored. We examine whether climate affects sovereign financing conditions, the extent of foreign risk transmission, and the relative magnitude of local versus global channels. Our primary contribution lies in mapping and quantifying climate spillovers through a novel measure that systematically tracks international value added linkages from Inter-Country Input-Output tables. We exploit exogenous variation in daily temperature, precipitation, and drought data weighted by gridded economic activity to attribute changes in financing conditions, measured by sovereign credit ratings, to domestic and foreign climate. Using standard and quantile regressions for 75 countries (2000-2022), we estimate average and heterogeneous effects and disentangle domestic versus spillover impacts. Results are threefold. First, local temperature anomalies and drought conditions exhibit significant negative and heterogeneous effects on sovereign ratings, with a one-unit increase implying an average 0.2 notch downgrade. Second, incorporating spillovers from foreign climate exposure increases estimated effects by 36% compared to local-only baselines, with countries in arid climate zones experiencing the largest impacts of up to one full notch. Third, for over 40% of countries, spillovers even exceed local impacts. For some small, highly globalized economies, mainly in Europe, spillovers contribute up to 80% of total climate impacts. Thus, ignoring global spillovers leads to a systematic misestimation of climate risk, revealing a critical blind spot for investors, regulators and policymakers.