  Recent work measuring the binary fraction of evolved red supergiants (RSGs)
in the Magellanic Clouds points to a value between 15-30%, with the majority of
the companions being un-evolved 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 lead to the identification of a
complete sample of RSGs down to a limiting $\log L/L_{\odot} \geq 4.2$. 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 ($41.2^{+12.0}_{-7.3}$%) than the outer regions
($15.9^{+12.4}_{-1.9}$%). Such a trend is not seen in M31; instead, the binary
fraction in lightly reddened regions remains constant at $33.5^{+8.6}_{-5.0}$%.
I conclude 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 changes in the parent distribution
of OB binaries.
