Climate models' outputs are affected by biases that need to be detected and adjusted to model climate impacts. Many climate hazards and climate-related impacts are associated with the interaction between multiple drivers, i.e. by compound events. So far climate model biases are typically assessed based on the hazard of interest, and it is unclear how much a potential bias in the dependence of the hazard drivers contributes to the overall bias and how the biases in the drivers interact. Here, based on copula theory, we develop a multivariate bias assessment framework, which allows for disentangling the biases in hazard indicators in terms of the underlying univariate drivers and their statistical dependence. Based on this framework, we dissect biases in fire and heat stress hazards in a suite of global climate models by considering two simplified hazard indicators, the wet-bulb globe temperature (WBGT) and the Chandler Burning Index (CBI). Both indices solely rely on temperature and relative humidity. The spatial pattern of the hazards indicators is well represented by climate models. However, substantial biases exist in the representation of extreme conditions, especially in the CBI (spatial average of absolute bias: 21 °C) due to the biases driven by relative humidity (20 °C). Biases in WBGT (1.1 °C) are small compared to the biases driven by temperature (1.9 °C) and relative humidity (1.4 °C), as the two biases compensate each other. In many regions, also biases related to the statistical dependence (0.85 °C) are important for WBGT, which indicates that well-designed physically-based multivariate bias adjustment should be considered for hazards and impacts that depend on multiple drivers. The proposed compound event-oriented evaluation of climate model biases is easily applicable to other hazard types. Furthermore, it can contribute to improved present and future risk assessments through increasing our understanding of the biases’ sources in the simulation of climate impacts.