This study employs the generalized extreme value - autoregressive conditional Fréchet - tail quotient correlation coefficient (GEV-AcF-TQCC) analytical framework to investigate the extreme co-movements between the EU carbon futures market and the commodity futures market. Subsequently, it also examines the response of the dynamic tail quotient correlation coefficient (TQCC) values to economic policy uncertainty stemming from the US using the linear Granger causality test. The empirical results demonstrate that the tail risk indices for both upside and downside trends of the two futures markets experience changes most of the time and show drastic changes during extreme events. More importantly, before the outbreaks of the COVID-19 pandemic (from September 2019 to February 2020) and the Russia-Ukraine conflict (from May 2021 to February 2022), the dynamic TQCC values for the downside trends of the two futures markets remain at relatively high levels. Of note, when examining the dynamic TQCC values between GSCI_down and EUA_down, as well as between GSCI_down and EUA_up, it can be observed that these TQCC values rise to higher levels during periods of extreme volatility. However, they quickly decline as the GSCI futures market experiences significant fluctuations during its downside trend. This pattern indicates a tail risk spillover effect from the GSCI futures market to the EUA futures market. Further analysis using the linear Granger causality test reveals that economic policy uncertainty is a significant driver of the extreme co-movements between the EU carbon and the commodity futures markets in most cases, in addition to the extreme co-movements between the downside trend of the EU carbon futures market and the upside trend of the commodity futures market. The insights gained from this study can provide valuable guidance and enlightenment for policymakers, financial investors, and academic researchers in understanding the dynamics of the extreme relationship between the two markets under discussion, and the impacts of economic policy uncertainty on their extreme co-movements.