  Geometric Invariant Theory (GIT) produces quotients of algebraic varieties by
reductive groups. If the variety is projective, this quotient depends on a
choice of polarisation; by work of Dolgachev-Hu and Thaddeus, it is known that
two quotients of the same variety using different polarisations are related by
birational transformations. Crucially, only finitely many birational varieties
arise in this way: variation of GIT fails to capture the entirety of the
birational geometry of GIT quotients. We construct a space parametrising all
possible GIT quotients of all birational models of the variety in a simple and
natural way, which captures the entirety of the birational geometry of GIT
quotients in a precise sense. It yields in particular a compactification of a
birational analogue of the set of stable orbits of the variety.
