  The coherent-constructible correspondence is a relationship between coherent
sheaves on a toric variety X, and constructible sheaves on a real torus T. This
was discovered by Bondal, and explored in the equivariant setting by Fang, Liu,
Treumann and Zaslow. In this paper we collect partial results towards a proof
of the non equivariant coherent-constructible correspondence. Also, we give
applications to the construction of tilting complexes in the derived category
of toric DM stacks.
