  We prove a theorem relating torus-equivariant coherent sheaves on toric
varieties to polyhedrally-constructible sheaves on a vector space. At the level
of K-theory, the theorem recovers Morelli's description of the K-theory of a
smooth projective toric variety. Specifically, let $X$ be a proper toric
variety of dimension $n$ and let $M_\bR = \mathrm{Lie}(T_\bR^\vee)\cong \bR^n$
be the Lie algebra of the compact dual (real) torus $T_\bR^\vee\cong U(1)^n$.
Then there is a corresponding conical Lagrangian $\Lambda \subset T^*M_\bR$ and
an equivalence of triangulated dg categories $\Perf_T(X) \cong
\Sh_{cc}(M_\bR;\Lambda),$ where $\Perf_T(X)$ is the triangulated dg category of
perfect complexes of torus-equivariant coherent sheaves on $X$ and
$\Sh_{cc}(M_\bR;\Lambda)$ is the triangulated dg category of complex of sheaves
on $M_\bR$ with compactly supported, constructible cohomology whose singular
support lies in $\Lambda$. This equivalence is monoidal---it intertwines the
tensor product of coherent sheaves on $X$ with the convolution product of
constructible sheaves on $M_\bR$.
