The observed stratospheric quasi‐biennial oscillation (QBO) and the tropospheric Madden‐Julian oscillation (MJO) are strongly connected in boreal winter, with stronger MJO activity when lower‐stratospheric winds are easterly. However, the current generation of climate models with internally generated representations of the QBO and MJO do not simulate the observed QBO‐MJO connection, for reasons that remain unclear. This study builds on prior work exploring the QBO‐MJO link in climate models whose stratospheric winds are relaxed toward reanalysis, reducing stratospheric biases in the model and imposing a realistic QBO. A series of ensemble experiments are performed using four state‐of‐the‐art climate models capable of representing the MJO over the period 1980–2015, each with similar nudging in the stratosphere. In these four models, nudging leads to a good representation of QBO wind and temperature signals, however no model simulates the observed QBO‐MJO relationship. Biases in MJO vertical structure and cloud‐radiative feedbacks are investigated, but no conclusive model bias or mechanism is identified that explains the lack of a QBO‐MJO connection.