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We demonstrate that highly accurate joint redshift-stellar mass probability distribution functions (PDFs) can be obtained using the Random Forest (RF) machine learning (ML) algorithm, even with few photometric bands available. As an example, we use the Dark Energy Survey (DES), combined with the COSMOS2015 catalogue for redshifts and stellar masses. We build two ML models: one containing deep photometry in the $griz$ bands, and the second reflecting the photometric scatter present in the main DES survey, with carefully constructed representative training data in each case. We validate our joint PDFs for $10,699$ test galaxies by utilizing the copula probability integral transform and the Kendall distribution function, and their univariate counterparts to validate the marginals. Benchmarked against a basic set-up of the template-fitting code BAGPIPES, our ML-based method outperforms template fitting on all of our predefined performance metrics. In addition to accuracy, the RF is extremely fast, able to compute joint PDFs for a million galaxies in just under $6$ min with consumer computer hardware. Such speed enables PDFs to be derived in real time within analysis codes, solving potential storage issues. As part of this work we have developed GALPRO, a highly intuitive and efficient Python package to rapidly generate multivariate PDFs on-the-fly. GALPRO is documented and available for researchers to use in their cosmology and galaxy evolution studies.