  We initiate an exploration of on-shell functions in $\mathcal{N}=4$ SYM
beyond the planar limit by providing compact, combinatorial expressions for all
leading singularities of MHV amplitudes and showing that they can always be
expressed as a positive sum of differently ordered Parke-Taylor tree
amplitudes. This is understood in terms of an extended notion of positivity in
$G(2,n)$, the Grassmannian of 2-planes in $n$ dimensions: a single on-shell
diagram can be associated with many different "positive" regions, of which the
familiar positive region associated with planar diagrams is just one example.
The decomposition into Parke-Taylor factors is simply a "triangulation" of
these extended positive regions. The $U(1)$ decoupling and Kleiss-Kuijf (KK)
relations satisfied by the Parke-Taylor amplitudes also follow naturally from
this geometric picture. These results suggest that non-planar MHV amplitudes in
$\mathcal{N}=4$ SYM at all loop orders can be expressed as a sum of
polylogarithms weighted by color factors and (unordered) Parke-Taylor
amplitudes.
