Let G be a reductive complex Lie group acting holomorphically on normal Stein spaces X and Y , which are locally G-biholomorphic over a common categorical quotient Q. When is there a global G-biholomorphism X ! Y ?If the actions of G on X and Y are what we, with justification, call generic, we prove that the obstruction to solving this local-to-global problem is topological and provide sufficient conditions for it to vanish. Our main tool is the equivariant version of Grauert's Oka principle due to Heinzner and Kutzschebauch.We prove that X and Y are G-biholomorphic if X is K-contractible, where K is a maximal compact subgroup of G, or if X and Y are smooth and there is a G-diffeomorphism W X ! Y over Q, which is holomorphic when restricted to each fibre of the quotient map X ! Q. We prove a similar theorem when is only a G-homeomorphism, but with an assumption about its action on G-finite functions. When G is abelian, we obtain stronger theorems. Our results can be interpreted as instances of the Oka principle for sections of the sheaf of G-biholomorphisms from X to Y over Q. This sheaf can be badly singular, even for a low-dimensional representation of SL 2 .C/.Our work is in part motivated by the linearisation problem for actions on C n . It follows from one of our main results that a holomorphic G-action on C n , which is locally G-biholomorphic over a common quotient to a generic linear action, is linearisable.