  Scattering amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory
exhibit singularities which reflect various aspects of the cluster algebras
associated to the Grassmannians ${\rm Gr}(4,n)$ and their tropical
counterparts. Here we investigate the potential origins of such structures and
examine the extent to which they can be recovered from the Gr\"obner structure
of the underlying Pl\"ucker ideals, focussing on the Grassmannians
corresponding to finite cluster algebras.
  Starting from the Pl\"ucker ideal, we describe how the polynomial cluster
variables are encoded in non-prime initial ideals associated to certain maximal
cones of the positive tropical fan. Following [1] we show that extending the
Pl\"ucker ideal by such variables leads to a Gr\"obner fan with a single
maximal Gr\"obner cone spanned by the positive tropical rays. The associated
initial ideal encodes the compatibility relations among the full set of cluster
variables. Thus we find that the Gr\"obner structure naturally encodes both the
symbol alphabet and the cluster adjacency relations exhibited by scattering
amplitudes without invoking the cluster algebra at all.
  As a potential application of these ideas we then examine the kinematic ideal
associated to non-dual conformal massless scattering written in terms of spinor
helicity variables. For five-particle scattering we find that the ideal can be
identified with the Pl\"ucker ideal for ${\rm Gr}(3,6)$ and the corresponding
tropical fan contains a number of non-prime ideals which encode all additional
letters of the two-loop pentagon function alphabet present in various
calculations of massless five-point finite remainders.
