Inspired by the BCFW recurrence for tilings of the amplituhedron, we introduce the general framework of `plabic tangles' that utilizes plabic graphs to define rational maps called `promotions' between products of Grassmannians. Our central conjecture is that promotion maps are quasi-cluster homomorphisms, which we prove for several classes of promotions. We describe the underlying operad structure, and relate promotion maps to vector-relation configurations and to the degree of the amplituhedron map on positroid varieties. Finally we give an example (the `4-mass box') which points to new positivity properties for non-rational maps beyond cluster algebras. This is based on joint work with Chaim Even-Zohar, Matteo Parisi, Melissa Sherman-Bennett, and Ran Tessler.