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We report the detection of a bright fast radio burst, FRB\,191108, with Apertif on the Westerbork Synthesis Radio Telescope (WSRT). The interferometer allows us to localise the FRB to a narrow $5\arcsec\times7\arcmin$ ellipse by employing both multibeam information within the Apertif phased-array feed (PAF) beam pattern, and across different tied-array beams. The resulting sight line 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 of M31, the Andromeda Galaxy. We find that the shared plasma of the Local Group galaxies could contribute $\sim$10\% of its dispersion measure of 588\,pc\,cm$^{-3}$. FRB\,191108 has a Faraday rotation measure of +474\,$\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 magnetised 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 $Δν\approx40$\,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.