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Baryon acoustic oscillations (BAOs) are a powerful probe of the expansion history of our Universe and are typically measured in the two-point statistics of a galaxy survey, either in Fourier space or in configuration space. In this work, we report a first measurement of BAOs from a joint fit of power spectrum and correlation function multipoles. We tested our new framework with a set of 1000 mock catalogs and showed that our method yields smaller biases on BAO parameters than individually fitting power spectra or correlation functions, or when combining them with the Gaussian approximation method. Our estimated uncertainties are slightly larger than those from the Gaussian approximation, likely due to noise in our sample covariance matrix, the larger number of nuisance parameters, or the fact that our new framework does not rely on the assumption of Gaussian likelihoods for the BAO parameters. However, we argue that our uncertainties are more reliable since they rely on fewer assumptions, and because our method takes correlations between Fourier and configuration space at the level of the two-point statistics. We performed a joint analysis of the luminous red galaxy sample of the extended baryon oscillation spectroscopic survey (eBOSS) data release 16, obtaining $D_H/r_d = 19.27 \pm 0.48$ and $D_M/r_d = 17.77 \pm 0.37$, in excellent agreement with the official eBOSS consensus BAO-only results $D_H/r_d = 19.33 \pm 0.53$ and $D_M/r_d =17.86 \pm 0.33$.