  We study tropical Grassmanians Tr$(k,n)$ in relation to cluster algebras, and
assess their applicability to $n$-particle amplitudes for $n=7,8$. In
$\mathcal{N}=4$ super Yang-Mills theory, we first show that while the totally
positive part of Tr$(4,7)$ may encompass the iterated discontinuity structure
of the seven-point Maximally Helicity Violating (MHV) amplitude, it is too
small for the Next-to-MHV helicity configuration. Then, using Tr$(4,8)$ we
propose a finite set of 356 cluster $\mathcal{A}$-coordinates expected to
contain the rational symbol letters of the eight-particle MHV amplitude, and
discuss how the remaining square-root letters may be obtained from limits of
infinite mutation sequences. Finally, we use a triangulation of the totally
positive part of Tr$(3,8)$ to obtain the associated generalised biadjoint
scalar amplitude in a form containing a near-minimal amount of spurious poles.
