Off-shell (4,4) supersymmetric sigma models with torsion as gauge theories in harmonic superspace

We present a manifestly supersymmetric off-shell formulation of a wide class of (4,4) 2D sigma models with torsion and noncommuting left and right complex structures in the harmonic superspace with a double set of SU(2) harmonic variables. The distinguishing features of the relevant superfield action 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 noncommutativity of complex stuctures on the bosonic target.