An approach to achieve refractive index sensing at an air and aqueous glycerol solution interface is proposed using a tapered-fiber-based microfiber Mach-Zehnder interferometer (MFMZI). Compared to a surrounding uniform medium of air or solutions, the spectral interference visibility of the MFMZI at the air/solution interface is significantly reduced due to a weak coupling between the fundamental cladding mode and high-order asymmetric cladding modes, which are extremely sensitive to the external refractive index. The MFMZI is experimentally demonstrated as an evanescent wave refractive index sensor to measure concentrations of glycerol solutions by monitoring average power attenuation of the tapered fiber.