  We prove a parametric Oka principle for equivariant sections of a holomorphic
fibre bundle $E$ with a structure group bundle $\mathscr G$ on a reduced Stein
space $X$, such that the fibre of $E$ is a homogeneous space of the fibre of
$\mathscr G$, with the complexification $K^\mathbb C$ of a compact real Lie
group $K$ acting on $X$, $\mathscr G$, and $E$. Our main result is that the
inclusion of the space of $K^\mathbb C$-equivariant holomorphic sections of $E$
over $X$ into the space of $K$-equivariant continuous sections is a weak
homotopy equivalence. The result has a wide scope; we describe several diverse
special cases. We use the result to strengthen Heinzner and Kutzschebauch's
classification of equivariant principal bundles, and to strengthen an Oka
principle for equivariant isomorphisms proved by us in a previous paper.
