We report the detection of a bright fast radio burst, FRB 191108, with Apertif on the Westerbork Synthesis Radio Telescope. The interferometer allows us to localize the FRB to a narrow 5 arcsec × 7 arcmin ellipse by employing both multibeam information within the Apertif phased-array feed beam pattern, and across different tied-array beams. The resulting sightline passes close to Local Group galaxy M33, with an impact parameter of only 18 kpc with respect to the core. It also traverses the much larger circumgalactic medium (CGM) of M31, the Andromeda Galaxy. We find that the shared plasma of the Local Group galaxies could contribute ∼10 per cent of its dispersion measure of 588 pc cm-3. FRB 191108 has a Faraday rotation measure (RM) of +474 $\pm \, 3$ rad m-2, which is too large to be explained by either the Milky Way or the intergalactic medium. Based on the more moderate RMs of other extragalactic sources that traverse the halo of M33, we conclude that the dense magnetized plasma resides in the host galaxy. The FRB exhibits frequency structure on two scales, one that is consistent with quenched Galactic scintillation and broader spectral structure with Δν ≍ 40 MHz. If the latter is due to scattering in the shared M33/M31 CGM, our results constrain the Local Group plasma environment. We found no accompanying persistent radio sources in the Apertif imaging survey data.