We present an analysis of 168 oscillating red giants from NASA's Kepler mission that exhibit anomalous peaks in their Fourier amplitude spectra. These peaks result from ellipsoidal variations that are indicative of binary star systems, at frequencies such that the orbit of any stellar companion would be within the convective envelope of the red giant. Alternatively, the observed phenomenon may be due to a close binary orbiting a red giant in a triple system, or chance alignments of foreground or background binary systems contaminating the target pixel aperture. We identify 87 stars in the sample as chance alignments using a combination of pixel Fourier analysis and difference imaging. We find that in the remaining 81 cases, the anomalous peaks are indistinguishable from the target star to within 4 arcsec, suggesting a physical association. We examine a GALAXIA model of the Kepler field of view to estimate background star counts and find that it is highly unlikely that all targets can be explained by chance alignments. From this, we conclude that these stars may comprise a population of physically associated systems.