  We prove that small, semi-linear Hamiltonian perturbations of the defocusing
nonlinear Schr\"odinger (dNLS) equation on the circle have an abundance of
invariant tori of any size and (finite) dimension which support quasi-periodic
solutions. When compared with previous results the novelty consists in
considering perturbations which do not satisfy any symmetry condition (they may
depend on $x$ in an arbitrary way) and need not be analytic. The main
difficulty is posed by pairs of almost resonant dNLS frequencies. The proof is
based on the integrability of the dNLS equation, in particular the fact that
the nonlinear part of the Birkhoff coordinates is one smoothing. We implement a
Newton-Nash-Moser iteration scheme to construct the invariant tori. The key
point is the reduction of linearized operators, coming up in the iteration
scheme, to $ 2 \times 2 $ block diagonal ones with constant coefficients
together with sharp asymptotic estimates of their eigenvalues.
