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We present photometric and spectroscopic data for the nearby Type I supernova (SN Ia) 2019eix (originally classified as an SN Ic), from the day of its discovery up to 100 days after maximum brightness. Before maximum light, SN 2019eix resembles a typical SN Ic, albeit lacking the usual O i feature. Its light curve is similar to the typical SN Ic with decline rates (Δ M _15, _V = 0.84) and absolute magnitude M _V = −18.35. However, after maximum light, this SN has unusual spectroscopic features, a large degree of line blending, significant line blanketing in the blue ( λ < 5000 Å), and strong Ca ii absorption features during and after peak brightness. These unusual spectral features are similar to models of subluminous thermonuclear explosions, specifically double-detonation models of SNe Ia. Photometrically, SN 2019eix appears to be somewhat brighter with slower decline rates than other double-detonation candidates. We modeled the spectra using the radiative-transfer code TARDIS using SN 1994I (an SN Ic) as a base model to see whether we could reproduce the unusual features of SN 2019eix and found them to be consistent with the exception of the O i feature. We also compared SN 2019eix with double-detonation models and found them to best match the observations of SN 2019eix, but failed to reproduce its full photometric and spectroscopic evolution.