Exotic full factors via weakly coarse bimodules

We are able to explicitly compute the bimodule structure of von Neumann algebra inclusions in handle constructions, which arise as inductive limits of iterated amalgamated free products not elementarily equivalent to $L(\mathbb{F}_2)$. Our computation is achieved via identifying delicate normal form decompositions in amalgamated free products built in an iterated fashion. Using these techniques, we are able to show that the handles constructions are always full, without any need to appeal to Property (T) phenomena which was essential in all previous works. Furthermore our bimodule machinery works in the setting of arbitrary von Neumann algebras equipped with faithful normal states, yielding examples of full $\mathrm{III}_1$ factors via handle constructions.