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In this article, we present a massively parallel framework for computingtropicalizations of algebraic varieties which can make use of finitesymmetries. We compute the tropical Grassmannian TGr$_0(3,8)$, and show that itrefines the $15$-dimensional skeleton of the Dressian Dr$(3,8)$ with theexception of $23$ special cones for which we construct explicit obstructions tothe realizability of their tropical linear spaces. Moreover, we proposealgorithms for identifying maximal-dimensional tropical cones which belong tothe positive tropicalization. These algorithms exploit symmetries of thetropical variety even though the positive tropicalization need not besymmetric. We compute the maximal-dimensional cones of the positiveGrassmannian TGr$^+(3,8)$ and compare them to the cluster complex of theclassical Grassmannian Gr$(3,8)$.

Parallel Computation of tropical varieties, their positive part, and tropical Grassmannians