We report on the first simultaneous XMM–Newton, NuSTAR, and Swift observations of the transitional millisecond pulsar PSR J1023+0038 in the X-ray active state. Our multi-wavelength campaign allowed us to investigate with unprecedented detail possible spectral variability over a broad energy range in the X-rays, as well as correlations and lags among emissions in different bands. The soft and hard X-ray emissions are significantly correlated, with no lags between the two bands. On the other hand, the X-ray emission does not correlate with the UV emission. We refine our model for the observed mode switching in terms of rapid transitions between a weak propeller regime and a rotation-powered radio pulsar state, and report on a detailed high-resolution X-ray spectroscopy using all XMM–Newton Reflection Grating Spectrometer data acquired since 2013. We discuss our results in the context of the recent discoveries on the system and of the state of the art simulations on transitional millisecond pulsars, and show how the properties of the narrow emission lines in the soft X-ray spectrum are consistent with an origin within the accretion disc.