We present a setup dedicated to the measurement of the small scalar directional anisotropies associated to the magnetochiral interaction. The apparatus, based on a polarization-independent fiber Sagnac interferometer, is optimized to be insensitive to circular anisotropies and to residual absorption. It can thus characterize samples of biological interests, for which the two enantiomers are not available and/or which present poor transmission. The signal-to-noise ratio is shown to be limited only by the source intensity noise, leading to a detection limit of Delta phi = 500 nrad Hz(-1/2). It yields a limit on the magnetochiral index n(MC) < 4 x 10(-13) T-1 at 1.55 mu m for the organic molecules tested. (C) 2014 AIP Publishing LLC.