Recent work measuring the binary fraction of evolved red supergiants (RSGs) in the Magellanic Clouds points to a value between 15% and 30%, with the majority of the companions being unevolved B-type stars as dictated by stellar evolution. Here I extend this research to the Local Group galaxies M31 and M33 and investigate the RSG binary fraction as a function of metallicity. Recent near-IR photometric surveys of M31 and M33 have led to the identification of a complete sample of RSGs down to a limiting logL/L{sun}>=4.2. To determine the binary fraction of these M31 and M33 RSGs, I used a combination of newly obtained spectroscopy to identify single RSGs and RSG+OB binaries, as well as archival UV, visible, and near-IR photometry to probabilistically classify RSGs as either single or binary based on their colors. I then adjusted the observed RSG+OB binary fraction to account for observational biases. The resulting RSG binary fraction in M33 shows a strong dependence on galactocentric distance, with the inner regions having a much higher binary fraction (41.2_-7.3_^+12.0^%) than the outer regions (15.9_-1.9_^+12.4^%). Such a trend is not seen in M31; instead, the binary fraction in lightly reddened regions remains constant at 33.5_-5.0_^+8.6^% . I conclude that the changing RSG binary fraction in M33 is due to a metallicity dependence, with higher-metallicity environments having higher RSG binary fractions. This dependence most likely stems not from changes in the physical properties of RSGs due to metallicity but from changes in the parent distribution of OB binaries.