  We continue the exploration of various appearances of cluster algebras in
scattering amplitudes and related topics in physics. The cluster configuration
spaces generalize the familiar moduli space ${\mathcal M}_{0,n}$ to finite-type
cluster algebras. We study worldsheet-like variables, which for classical types
have also appeared in the study of the symbol alphabet of Feynman integrals. We
provide a systematic derivation of these variables from $Y$-systems, which
allows us to express the dihedral coordinates in terms of them and to write the
corresponding cluster string integrals in compact forms. We mainly focus on the
$D_n$ type and show how to reach the boundaries of the configuration space, and
write the saddle-point equations in terms of these variables. Moreover, these
variables make it easier to study various topological properties of the space
using a finite-field method. We propose conjectures about quasi-polynomial
point count, dimensions of cohomology, and the number of saddle points for the
$D_n$ space up to $n=10$, which greatly extend earlier results.
