  We present the Sherwood-Relics simulations, a new suite of large cosmological
hydrodynamical simulations aimed at modelling the intergalactic medium (IGM)
during and after the cosmic reionization of hydrogen. The suite consists of
over 200 simulations that cover a wide range of astrophysical and cosmological
parameters. It also includes simulations that use a new lightweight hybrid
scheme for treating radiative transfer effects. This scheme follows the spatial
variations in the ionizing radiation field, as well as the associated
fluctuations in IGM temperature and pressure smoothing. It is computationally
much cheaper than full radiation hydrodynamics simulations and circumvents the
difficult task of calibrating a galaxy formation model to observational
constraints on cosmic reionization. Using this hybrid technique, we study the
spatial fluctuations in IGM properties that are seeded by patchy cosmic
reionization. We investigate the relevant physical processes and assess their
impact on the z > 4 Lyman-alpha forest. Our main findings are: (i) Consistent
with previous studies patchy reionization causes large scale temperature
fluctuations that persist well after the end of reionization, (ii) these
increase the Lyman-alpha forest flux power spectrum on large scales, and (iii)
result in a spatially varying pressure smoothing that correlates well with the
local reionization redshift. (iv) Structures evaporated or puffed up by
photoheating cause notable features in the Lyman-alpha forest, such as
flat-bottom or double-dip absorption profiles.
