The cycling of carbon in the oceans is affected by feedbacks driven by changes in climate and atmospheric CO2. Understanding these feedbacks is therefore an important prerequisite for projecting future climate. Marine biogeochemistry models are a useful tool but, as with any model, are a simplification and need to be continually improved. In this study, we coupled the Finite-volumE Sea ice–Ocean Model (FESOM2.1) to the Regulated Ecosystem Model version 3 (REcoM3). FESOM2.1 is an update of the Finite-Element Sea ice–Ocean Model (FESOM1.4) and operates on unstructured meshes. Unlike standard structured-mesh ocean models, the mesh flexibility allows for a realistic representation of small-scale dynamics in key regions at an affordable computational cost. Compared to the previous coupled model version of FESOM1.4–REcoM2, the model FESOM2.1–REcoM3 utilizes a new dynamical core, based on a finite-volume discretization instead of finite elements, and retains central parts of the biogeochemistry model. As a new feature, carbonate chemistry, including water vapour correction, is computed by mocsy 2.0. Moreover, REcoM3 has an extended food web that includes macrozooplankton and fast-sinking detritus. Dissolved oxygen is also added as a new tracer. In this study, we assess the ocean and biogeochemical state simulated with FESOM2.1–REcoM3 in a global set-up at relatively low spatial resolution forced with JRA55-do (Tsujino et al., 2018) atmospheric reanalysis. The focus is on the recent period (1958–2021) to assess how well the model can be used for present-day and future climate change scenarios on decadal to centennial timescales. A bias in the global ocean–atmosphere preindustrial CO2 flux present in the previous model version (FESOM1.4–REcoM2) could be significantly reduced. In addition, the computational efficiency is 2–3 times higher than that of FESOM1.4–REcoM2. Overall, it is found that FESOM2.1–REcoM3 is a skilful tool for ocean biogeochemical modelling applications.