General relativity can be constructed as a gauge theory using the quotient manifold strategy of [1, 2]. We consider a conformal gauging where the geometry is far richer than normal spacetime, including a symplectic form and the necessary emergence of Lorentzian signature. The resulting 2n-dim manifold constitutes a relativistic phase space, and general relativity is recovered when we demand that the momentum space is flat. However, the full geometry allows for curved phase space.