  We study metric spaces defined via a conformal weight, or more generally a
measurable Finsler structure, on a domain $\Omega \subset \mathbb{R}^2$ that
vanishes on a compact set $E \subset \Omega$ and satisfies mild assumptions.
Our main question is to determine when such a space is quasiconformally
equivalent to a planar domain. We give a characterization in terms of the
notion of planar sets that are removable for conformal mappings. We also study
the question of when a quasiconformal mapping can be factored as a
1-quasiconformal mapping precomposed with a bi-Lipschitz map.
