  Starting from the dual action of $(4,4)$ $2D$ twisted multiplets in the
harmonic superspace with two independent sets of $SU(2)$ harmonic variables, we
present its generalization which hopefully provides an off-shell description of
general $(4,4)$ supersymmetric sigma models with torsion. Like the action of
the torsionless $(4,4)$ hyper-K\"ahler sigma models in the standard harmonic
superspace, it is characterized by a number of superfield potentials. They
depend on $n$ copies of a triple of analytic harmonic $(4,4)$ superfields. As
distinct from the hyper-K\"ahler case, the potentials prove to be severely
constrained by the self-consistency condition which stems from the
commutativity of the left and right harmonic derivatives. We show that for
$n=1$ these constraints reduce the general action to that of $(4,4)$ twisted
multiplet, while for $n\geq 2$ there exists a wide class of new actions which
cannot be written only via twisted multiplets. Their most striking feature is
the nonabelian and in general nonlinear gauge invariance which substitutes the
abelian gauge symmetry of the dual action of twisted multiplets and ensures the
correct number of physical degrees of freedom. We show, on a simple example,
that these actions describe sigma models with non-commuting left and right
complex structures on the bosonic target.
