Demazure weaves are combinatorial representations of maps between Bott--Samelson cells and have been used to construct cluster structures on braid varieties. We show the compatibility of these maps and the resulting cluster structures with the standard Poisson structure on the Bott--Samelson variety. Adding frayed strands to Demazure weaves, we further construct Poisson compatible cluster structures on other affine charts of the Bott--Samelson variety in a manner that transition functions across charts become rational quasi-cluster. The mutation sequences we construct for these quasi-cluster morphisms are closely related to those of Ménard for open Richardson seeds.