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I think this should work: The Lie algebra action extends, by multiplying by $i$. By compactness, these vector fields are all complete. So some covering group of the complexification acts. The fundamental group embeds into the universal covering group, and we need to see which elements of it act trivially. So if the fundamental group of the complexification is the same as that of the original compact group, i.e. if the original compact group is a maximal compact subgroup, then that fundamental group acts trivially. Hence the complexification acts on the complex homogeneous space as long as the compact injects into the complexification.