The binary fraction of unevolved massive stars is thought to be 70%-100% but there are few observational constraints on the binary fraction of the evolved version of a subset of these stars, the red supergiants (RSGs). Here we identify a complete sample of RSGs in the Large Magellanic Cloud (LMC) using new spectroscopic observations and archival UV, IR, and broadband optical photometry. We find 4090 RSGs with logL/L_{sun}_>3.5, with 1820 of them having logL/L_{sun}_>4, which we believe is our completeness limit. We additionally spectroscopically confirmed 38 new RSG + B-star binaries in the LMC, bringing the total known up to 55. We then estimated the binary fraction using a k-nearest neighbors algorithm that classifies stars as single or binary based on photometry with a spectroscopic sample as a training set. We take into account observational biases such as line-of-sight stars and binaries in eclipse while also calculating model- dependent corrections for RSGs with companions that our observations were not designed to detect. Based on our data, we find an initial result of 13.5_-6.67_^+7.56^% for RSGs with O- or B-type companions. Using the Binary Population and Spectral Synthesis models to correct for unobserved systems, this corresponds to a total RSG binary fraction of 19.5_-6.7_^+7.6^% . This number is in broad agreement with what we would expect given an initial OB binary distribution of 70%, a predicted merger fraction of 20%-30%, and a binary interaction fraction of 40%-50%.