  We describe the multi-moment map associated to an almost Hermitian manifold
which admits an action of a torus by holomorphic isometries. We investigate in
particular the case of a $\mathbb T^3$ action on the homogeneous nearly
K\"ahler $ S^3\times S^3$. We find that the multi-moment map in this case acts
more-or-less similarly to the moment map of a toric manifold, while the more
general case does not.
