  Starting from the action of $(4,4)$ $2D$ twisted multiplets in the harmonic
superspace with a double set of $SU(2)$ harmonic variables, we present its
generalization which provides an off-shell description of a wide class of
$(4,4)$ sigma models with torsion and non-commuting left and right complex
structures. The distinguishing features of the action constructed are: (i) a
nonabelian and in general nonlinear gauge invariance ensuring a correct number
of physical degrees of freedom; (ii) an infinite tower of auxiliary fields. For
a particular class of such models we explicitly demonstrate the
non-commutativity of complex structures on the bosonic target.
