Homological mirror symmetry statements have long been sought and proposed for both positroid varieties in the Grassmannian and cluster varieties. Working towards the former and furnishing many examples of the latter, we prove HMS for any really full rank acyclic cluster variety. In particular this proves mirror symmetry for all finite-type positroid cells, and we issue open HMS conjectures relating this theorem to expected results for the corresponding positroid varieties.

Our main technical tool is a generalization of Gross-Hacking-Keel's construction of truncated cluster varieties to a class of spaces, called partially truncated cluster varieties (PTCV's), which contains all acyclic cluster varieties. In particular our proof establishes and relies on a version of FLTZ-Kuwagaki's and Gammage-Shende's naturality of toric mirror symmetry with respect to toric embeddings for certain embeddings of PTCV's.