Cloud phase improvements in a state‐of‐the‐art climate model produce a large 1.5 K increase in equilibrium climate sensitivity (ECS, the surface warming in response to instantaneously doubled CO2) via extratropical shortwave cloud feedbacks. Here we show that the same model improvements produce only a small surface warming increase in a realistic 21st-century emissions scenario. The small 21st-century warming increase is attributed to extratropical ocean heat uptake. Southern Ocean mean‐state circulation takes up heat while a slowdown in North Atlantic circulation acts as a feedback to slow surface warming. Persistent heat uptake by extratropical oceans implies that extratropical cloud biases may not be as important to 21st-century warming as biases in other regions. Observational constraints on cloud phase and shortwave radiation that produce a large ECS increase do not imply large changes in 21st-century warming.