  We address the impact of asymmetric nonlocal diffusion nonlinearity on the
properties of gap solitons supported by photorefractive crystal with an
imprinted optical lattice. We reveal how the asymmetric nonlocal response
alters the domains of existence and the stability of solitons originating from
different gaps. We find that in such media gap solitons cease to exist above a
threshold of the nonlocality degree. We discuss how the interplay between
nonlocality and lattice strength modifies the gap soliton mobility.
