Liquid xenon (LXe) is employed in a number of current and future detectors for rare event searches. We use the EXO-200 experimental data to measure the absolute scintillation and ionization yields generated by $gamma$ interactions from $^{228}$Th (2615~keV), $^{226}$Ra (1764~keV) and $^{60}$Co (1332~keV and 1173~keV) calibration sources, over a range of electric fields. The $W$-value that defines the recombination-independent energy scale is measured to be $11.5~pm~0.5$~(syst.)~$pm~0.1$~(stat.) eV. These data are also used to measure the recombination fluctuations in the number of electrons and photons produced by the calibration sources at the MeV-scale, which deviate from extrapolations of lower-energy data. Additionally, a semi-empirical model for the energy resolution of the detector is developed, which is used to constrain the recombination efficiency, i.e., the fraction of recombined electrons that result in the emission of a detectable photon. Detailed measurements of the absolute charge and light yields for MeV-scale electron recoils are important for predicting the performance of future neutrinoless double beta decay detectors.
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