Modelling the climate system on multi-millennial timescales is challenging when slow-response components, such as the deep ocean, vegetation, and ice sheets, must evolve alongside fast-response components such as atmospheric weather systems. This is crucial for investigating, for example, the dynamical structure of Earth's climate, including steady states, mapping the attractor of those states in a multi-dimensional phase space, and their response to external forcing and internal variability. Earth system models, such as those used in the Coupled Model Intercomparison Project (CMIP), are often too computationally expensive for simulations spanning many thousands of years. Moreover, simplified parameterizations and coarse resolutions typically employed in Earth Models of Intermediate Complexity (EMICs) can adversely affect the nonlinear interactions among the various climate components. Here, we describe a new tool, Biogeodynamics-Ice sheet-Geneva-MITgcm – or BIG-MITgcm for short – which attempts to fill in the hierarchy between these two classes of model. The core of BIG-MITgcm is a coupled MITgcm setup that includes atmospheric, ocean, thermodynamic sea ice, and land modules. To this, we asynchronously couple a vegetation model (BIOME4), a hydrological model (pysheds), and a new global-scale ice sheet model (MITgcmIS). The latter is implemented on the same cubed-sphere grid as MITgcm, using the shallow-ice approximation, and driven by a modified Positive Degree Day method to evaluate the ice-sheet surface mass balance. Here, we present a detailed description of the new ice sheet model and the coupling procedure employed. We evaluate BIG-MITgcm using a pre-industrial simulation initialized from observations of bedrock topography, together with a forced simulation over the 1979–2009 period. The model spontaneously grows plausible ice sheets. These two experiments allow us to assess the model's performance against CMIP-class models, as well as a combination of reanalyses and observations. To evaluate the ability of our model to represent completely different climate conditions and continental configurations, we also discuss a Permian-Triassic solution with a small ice sheet in the Northern Hemisphere. In summary, BIG-MITgcm successfully captures many large-scale properties of the current climate, suggesting that it will be a very useful tool for exploring current, past, and future climates. We conclude by discussing potential applications and future developments.