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Cosmological constraints derived from weak lensing (WL) surveys are limited by baryonic effects, which suppress the non-linear matter power spectrum on small scales. By combining WL measurements with data from external tracers of the gas around massive structures, it is possible to calibrate baryonic effects and, therefore, obtain more precise cosmological constraints. In this study, we generate mock data for a Stage-IV weak lensing survey such as the Legacy Survey of Space and Time (LSST), X-ray gas fractions, and stacked kinetic Sunyaev-Zel'dovich (kSZ) measurements, to jointly constrain cosmological and astrophysical parameters describing baryonic effects (using the Baryon Correction Model - BCM). First, using WL data alone, we quantify the level to which the BCM parameters will need to be constrained to recover the cosmological constraints obtained under the assumption of perfect knowledge of baryonic feedback. We identify the most relevant baryonic parameters and determine that they must be calibrated to a precision of ∼10-20% to avoid significant degradation of the fiducial WL constraints. We forecast that long-term X-ray data from ∼5000 clusters should be able to reach this threshold for the parameters that characterise the abundance of hot virialised gas. Constraining the distribution of ejected gas presents a greater challenge, however, but we forecast that long-term kSZ data from a CMB-S4-like experiment should achieve the level of precision required for full self-calibration.