AbstractTropical forests play a key role in absorbing carbon from the atmosphere into the land surface. Recent analyses of long‐term (1985–2014) forest inventory plots across the tropics show that structurally intact tropical forest are a large carbon sink, but that this sink has saturated and is projected to be in long‐term decline. Here we compare these results with estimates from the two latest generations of Earth System Models, Climate Modelling Intercomparison Project 5 (CMIP5) (19 models) and CMIP6 (17 models). While CMIP5 and CMIP6 are of similar skill, they do not reproduce the observed 1985–2014 carbon dynamics. The “natural” pan‐tropical carbon sink from inventory data is 0.99 Pg C yr⁻¹ (95% CI 0.7–1.3, n = 614) between 2000 and 2010, the best sampled decade, double the CMIP6 multimodel‐mean of 0.45 Pg C yr⁻¹ (95% CI 0.35–0.55). The observed saturating and declining sink is not captured by the models, which show modest increases in sink strength. The future (2015–2040) “natural” pan‐tropical sink from a statistical model driven by extrapolating past trends of its putative environmental drivers decreases by 0.23 Pg C per decade (95% CI 0.09–0.39) until the 2030s, while the CMIP6 multimodel‐mean under the climate change scenario closest to the statistical model project an increasing carbon sink (0.54 Pg C per decade; 95% CI 0.25–0.67). CMIP multimodel‐means reproduce the response of carbon gains from tree growth to environmental drivers, but the modeling of carbon losses from tree mortality does not correspond well to the inventory data. The model‐observation differences primarily result from the treatment of mortality in models.