Abstract We use the ‘Evolution and Assembly of GaLaxies and their Environments’ (eagle) suite of hydrodynamical cosmological simulations to measure offsets between the centres of stellar and dark matter components of galaxies. We find that the vast majority (>95 per cent) of the simulated galaxies display an offset smaller than the gravitational softening length of the simulations (Plummer-equivalent ϵ = 700 pc), both for field galaxies and satellites in clusters and groups. We also find no systematic trailing or leading of the dark matter along a galaxy's direction of motion. The offsets are consistent with being randomly drawn from a Maxwellian distribution with σ ≤ 196 pc. Since astrophysical effects produce no feasible analogues for the $1.62^{+0.47}_{-0.49}$ kpc offset recently observed in Abell 3827, the observational result is in tension with the collisionless cold dark matter model assumed in our simulations.