  The notion of crossed product by a coquasi-bialgebra H is introduced and
studied. The resulting crossed product is an algebra in the monoidal category
of right H-comodules. We give an interpretation of the crossed product as an
action of a monoidal category. In particular, necessary and sufficient
conditions for two crossed products to be equivalent are provided. Then, two
structure theorems for coquasi Hopf modules are given. First, these are
relative Hopf modules over the crossed product. Second, the category of
coquasi-Hopf modules is trivial, namely equivalent to the category of modules
over the starting associative algebra. In connection the crossed product, we
recall the notion of a cleft extension over a coquasi-Hopf algebra. A Morita
context of Hom spaces is constructed in order to explain these extensions,
which are shown to be equivalent with crossed product with invertible cocycle.
At the end, we give a complete description of all cleft extensions by the
non-trivial coquasi-Hopf algebras of dimension two and three.
