  The decomposition locus of a tensor is the set of rank-one tensors appearing
in a minimal tensor-rank decomposition of the tensor. For tensors lying on the
tangential variety of any Segre variety, but not on the variety itself, we show
that the decomposition locus consists of all rank-one tensors except the
tangency point only. We also explicitly compute decomposition loci of all
tensors belonging to tensor spaces with finitely many orbits with respect to
the action of product of general linear groups.
