  We show the existence of cluster $\mathcal{A}$-structures and cluster Poisson
structures on any braid variety, for any simple Lie group. The construction is
achieved via weave calculus and a tropicalization of Lusztig's coordinates.
Several explicit seeds are provided and the quiver and cluster variables are
readily computable. We prove that these upper cluster algebras equal their
cluster algebras, show local acyclicity, and explicitly determine their
DT-transformations as the twist automorphisms of braid varieties. The main
result also resolves the conjecture of B. Leclerc on the existence of cluster
algebra structures on the coordinate rings of open Richardson varieties.
