As a primary source of global Earth system predictability on subseasonal time scale, the Madden-Julian Oscillation (MJO) plays crucial roles in the global water cycle and has profound impacts on global climate and weather extremes. Despite progress made in the past, state-of-the-art global climate models, including DOE E3SM, still have great difficulty in realistically simulating MJOs. A two-moment five-class convective microphysics parameterization (CMP) was implemented in a version of E3SMv2, in which the P3 microphysics scheme was employed for stratiform clouds. It is shown that the E3SM with CMP simulates more realistic MJOs. Given that precipitation from grid-scale (stratiform) cloud microphysics contributes about half to the global annual mean precipitation in E3SM, we evaluated MJO simulation in convective and stratiform precipitation separately. It is found that E3SM can simulate the eastward propagation of the MJO in the convective precipitation component, although the propagation speed is overly fast. However, stratiform precipitation greatly degrades MJO simulation, showing westward propagation of precipitation over the Indian Ocean, very fast eastward propagation over the Maritime Continent, and no MJO signal in the western Pacific. The E3SM with CMP not only simulates the eastward propagation of the MJO at a realistic speed in convective precipitation but also improves MJO simulation in the stratiform precipitation component. Process-level sensitivity experiments suggest that the ice-phase microphysical processes play a key role in improving MJO simulation in convective precipitation. Further analysis indicates that CMP enhances the coupling between convection and stratiform clouds microphysics schemes through increased detrainment. In this way, the improved MJO signal in convection can be effectively transferred to the stratiform clouds microphysics scheme, improving the MJO simulation in stratiform precipitation. This analysis reveals that the interaction between convection and stratiform clouds is important for the realistic simulation of the MJO in E3SM.