Observations of local galaxies harbouring supermassive black holes (BHs) of anomalously high mass, M_BH, relative to their stellar mass, M_star, appear to be at odds with simple models of the co-evolution between galaxies and their central BHs. We study the origin of such outliers in a Lambda cold dark matter context using the EAGLE cosmological, hydrodynamical simulation. We find 15 M_BH(M_star)-outlier galaxies, defined as having M_BH more than 1.5 dex above the median M_BH(M_star) relation in the simulation, M_{BH,med}. All M_BH(M_star)-outliers are satellite galaxies, typically with M_star ~ 10^10 M_sun and M_BH ~ 10^8 M_sun. They have all become outliers due to a combination of tidal stripping of their outer stellar component acting over several Gyr and early formation times leading to rapid BH growth at high redshift, with the former mechanism being most important for 67 per cent of these outliers. The same mechanisms also cause the M_BH(M_star)-outlier satellites to be amongst the most compact galaxies in the simulation, making them ideal candidates for ultracompact dwarf galaxy progenitors. The 10 most extreme central galaxies found at z=0 (with log_{10}(M_BH/M_{BH,med}) in [1.2, 1.5]) grow rapidly in M_BH to lie well above the present-day M_BH-M_star relation at early times (z > 2), and either continue to evolve parallel to the z=0 relation or remain unchanged until the present day, making them relics of the high-redshift universe. This high-z formation mechanism may help to explain the origin of observed M_BH(M_star)-outliers with extended dark matter haloes and undisturbed morphologies.
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