  In this article, we present a massively parallel framework for computing
tropicalizations of algebraic varieties which can make use of finite
symmetries. We compute the tropical Grassmannian TGr$_0(3,8)$, and show that it
refines the $15$-dimensional skeleton of the Dressian Dr$(3,8)$ with the
exception of $23$ special cones for which we construct explicit obstructions to
the realizability of their tropical linear spaces. Moreover, we propose
algorithms for identifying maximal-dimensional tropical cones which belong to
the positive tropicalization. These algorithms exploit symmetries of the
tropical variety even though the positive tropicalization need not be
symmetric. We compute the maximal-dimensional cones of the positive
Grassmannian TGr$^+(3,8)$ and compare them to the cluster complex of the
classical Grassmannian Gr$(3,8)$.
