Addressing emissions of non-CO2 greenhouse gases (GHGs) is an integral part of efficient climate change mitigation and therefore an essential part of climate policy. Metrics are used to aggregate and compare emissions of short- and long-lived GHGs and need to account for the difference in both magnitude and persistence of their climatic effects. Different metrics describe different approaches and perspectives, and hence yield different numerical estimates for aggregated GHG emissions. When interpreting GHG emission reduction targets, being mindful of the underlying metrical choices thus proves to be essential. Here we present the impact a recently proposed GHG metric related to the concept of CO2 forcing-equivalent emissions (called GWP*) would have on the internal consistency and environmental integrity of the Paris Agreement. We show that interpreting the Paris Agreement goals in a metric like GWP* that is significantly different from the standard metric used in the IPCC Fifth Assessment Report can lead to profound inconsistencies in the mitigation architecture of the Agreement. It could even undermine the integrity of the Agreement's mitigation target altogether by failing to deliver net-zero CO2 emissions and therewith failing to ensure warming is halted. Our results indicate that great care needs to be taken when applying new concepts that appear scientifically favourable to a pre-existing climate policy context.