  We introduce quasi-homomorphisms of cluster algebras, a flexible notion of a
map between cluster algebras of the same type (but with different
coefficients). The definition is given in terms of seed orbits, the smallest
equivalence classes of seeds on which the mutation rules for non-normalized
seeds are unambiguous. We present examples of quasi-homomorphisms involving
familiar cluster algebras, such as cluster structures on Grassmannians, and
those associated with marked surfaces with boundary. We explore the related
notion of a quasi-automorphism, and compare the resulting group with other
groups of symmetries of cluster structures. For cluster algebras from surfaces,
we determine the subgroup of quasi-automorphisms inside the tagged mapping
class group of the surface.
