We investigate the alignment of galaxies and haloes relative to cosmic web filaments using the EAGLE hydrodynamical simulation. We identify filaments by applying the nexus+ method to the mass distribution and the Bisous formalism to the galaxy distribution. Both return similar filamentary structures that are well aligned and that contain comparable galaxy populations. EAGLE haloes have an identical spin alignment with filaments as their counterparts in dark-matter-only simulations: a complex mass-dependent trend with low-mass haloes spinning preferentially parallel to and high-mass haloes spinning preferentially perpendicular to filaments. In contrast, galaxy spins do not show such a transition and have a propensity for perpendicular alignments at all masses, with the degree of alignment being largest for massive galaxies. This result is valid for both nexus+ and Bisous filaments. When splitting by morphology, we find that elliptical galaxies show a stronger orthogonal spin–filament alignment than spiral galaxies of similar mass. The same is true of their host haloes. Due to the misalignment between galaxy shape and spin, galaxy minor axes are oriented differently with filaments than galaxy spins. We find that the galaxies whose minor axis is perpendicular to a filament are much better aligned with their host haloes. This suggests that many of the same physical processes determine both the galaxy–filament and the galaxy–halo alignments. The volume of the EAGLE simulation is relatively small and many of the alignments we have found are weak; validation of our conclusions will require hydrodynamical simulations of significantly larger volumes.