We propose a transparent climate debt index incorporating both methane (CH4) and carbon dioxide (CO2) emissions. We develop national historic emissions databases for both greenhouse gases to 2005, justifying 1950 as the starting point for global perspectives. We include CO2 emissions from fossil sources [CO2(f)], as well as, in a separate analysis, land use change and forestry. We calculate the CO2(f) and CH4 remaining in the atmosphere in 2005 from 205 countries using the Intergovernmental Panel on Climate Change's Fourth Assessment Report impulse response functions. We use these calculations to estimate the fraction of remaining global emissions due to each country, which is applied to total radiative forcing in 2005 to determine the combined climate debt from both greenhouse gases in units of milliwatts per square meter per country or microwatts per square meter per person, a metric we term international natural debt (IND). Australia becomes the most indebted large country per capita because of high CH4 emissions, overtaking the United States, which is highest for CO2(f). The differences between the INDs of developing and developed countries decline but remain large. We use IND to assess the relative reduction in IND from choosing between CO2(f) and CH4`control measures and to contrast the imposed versus experienced health impacts from climate change. Based on 2005 emissions, the same hypothetical impact on world 2050 IND could be achieved by decreasing CH4 emissions by 46% as stopping CO2 emissions entirely, but with substantial differences among countries, implying differential optimal strategies. Adding CH4 shifts the basic narrative about differential international accountability for climate change.