  We prove a version of the Chevalley Restriction Theorem for the action of a
real reductive group G on a topological space X which locally embeds into a
holomorphic representation. Assuming that there exists an appropriate quotient
X//G for the G-action, we introduce a stratification which is defined with
respect to orbit types of closed orbits. Our main result is a description of
the quotient X//G in terms of quotients by normalizer subgroups associated to
the stratification.
