We present results from a subset of simulations from the `Evolution and Assembly of GaLaxies and their Environments' (EAGLE) suite in which the formulation of the hydrodynamics scheme is varied. We compare simulations that use the same subgrid models without recalibration of the parameters but employing the standard GADGET flavour of smoothed particle hydrodynamics (SPH) instead of the more recent state-of-the-art ANARCHY formulation of SPH that was used in the fiducial EAGLE runs. We find that the properties of most galaxies, including their masses and sizes, are not significantly affected by the details of the hydrodynamics solver. However, the star formation rates of the most massive objects are affected by the lack of phase mixing due to spurious surface tension in the simulation using standard SPH. This affects the efficiency with which AGN activity can quench star formation in these galaxies and it also leads to differences in the intragroup medium that affect the X-ray emission from these objects. The differences that can be attributed to the hydrodynamics solver are, however, likely to be less important at lower resolution. We also find that the use of a time-step limiter is important for achieving the feedback efficiency required to match observations of the low-mass end of the galaxy stellar mass function.