  In this paper we explore the geometric space parametrized by (tree level)
Wilson loops in SYM $N=4$. We show that, this space can be seen as a vector
bundle over a totally non-negative subspace of the Grassmannian,
$\mathcal{W}_{k,cn}$. Furthermore, we explicitly show that this bundle is
non-orientable in the majority of the cases, and conjecture that it is
non-orientable in the remaining situation. Using the combinatorics of the
Deodhar decomposition of the Grassmannian, we identify subspaces $\Sigma(W)
\subset \mathcal{W}_{k,n}$ for which the restricted bundle lies outside the
positive Grassmannian. Finally, while probing the combinatorics of the Deodhar
decomposition, we give a diagrammatic algorithm for reading equations
determining each Deodhar component as a semialgebraic set.
