  We describe a family of tropical fans related to Grassmannian cluster
algebras. These fans are related to the kinematic space of massless scattering
processes in a number of ways. For each fan associated to the Grassmannian
${\rm Gr}(k,n)$ there is a notion of a generalised $\phi^3$ amplitude and an
associated set of scattering equations which further generalise the ${\rm
Gr}(k,n)$ scattering equations that have been recently introduced. Here we
focus mostly on the cases related to finite Grassmannian cluster algebras and
we explain how face variables for the cluster polytopes are simply related to
the scattering equations. For the Grassmannians ${\rm Gr}(4,n)$ the tropical
fans we describe are related to the singularities (or symbol letters) of loop
amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory. We show how each
choice of tropical fan leads to a natural class of polylogarithms, generalising
the notion of cluster adjacency and we describe how the currently known loop
data fit into this classification.
