$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\O{O}\DeclareMathOperator\Iso{Iso}$Let $ g $ be the round metric on the sphere $ S^n $. Since $ S^n $ is compact the isometry group $ \Iso(S^n,g) $ is also compact. And every compact group can be realized as the real points of some (reductive) linear algebraic group. Indeed, $ \Iso(S^n,g) = \O_{n+1}(\mathbb{R}) $. The complex points of this group are $ \O_{n+1}(\mathbb{C}) $. And $ \O_{n+1}(\mathbb{C}) $ acts transitively on the tangent bundle of the sphere $ T(S^n) $.
Does this generalize from the round sphere to other compact homogeneous Riemannian manifolds?
In other words, Let $ (M,g) $ be a compact Riemannian homogeneous space. Then $ \Iso(M,g) $ is a compact Lie group. So there exists some (reductive) linear algebraic group whose real points are isomorphic to $ \Iso(M,g) $. The question is, does there always exist a linear algebraic group $ G $ such that the real points of $ G $ are isomorphic to the isometry group
$$
G_\mathbb{R} \cong \Iso(M,g)
$$
and, in addition, the complex points of $ G $ act (transitively, smoothly) on the tangent bundle $ T(M) $?
Note that this question is equivalent to a question which does not a priori in involve any geometry:
The manifold
$
G_\mathbb{C}/H_\mathbb{C}
$
is the tangent bundle to
$
G_\mathbb{R}/H_\mathbb{R}
$
where $ G_\mathbb{R}$, $H_\mathbb{R} $ are compact real forms of $ G_\mathbb{C}$, $H_\mathbb{C} $
Note that while the action of $ G_\mathbb{R} $ is by isometries, the action of $ G_\mathbb{C} $ on $ T(M) $ can only be by isometries if $ M $ is parallelizable. So in particular the action of $ \O_{n+1}(\mathbb{C}) $ on $ T(S^n) $ can only be by isometries in the cases $ n=1,3,7 $.
Best Answer
Maybe these arguments are of interest to you. It is known that for any compact symmetric space $M$ the tangent bundle $TM$ possesses a canonical structure of a complex manifold. Multiplication by $-1$ on $TM$ is an antiholomorphic involution; its set of fixed points is $M$, when identified with the zero section of $TM$ (see e.g. Thm 2.5a of Szőke, R.: Complex structures on tangent bundles of Riemannian manifolds, Math. Ann. 291 (1991), 409-428). It follows that the action of the Lie group $G_{\mathbf R}$ on $M$ extends to an action on the complex manifold $TM$ by holomorphic automorphisms. One proves that it extends to a holomorphic action of the complexification $G_{\mathbf C}$ of $G_{\mathbf R}$ on $TM$. This can be done by considering the induced morphism from the real Lie algebra ${\mathfrak g}_{\mathbf R}$ of $G_{\mathbf R}$ into the complex Lie algebra ${\mathcal A}(TM)$ of holomorphic vector fields on $TM$. It naturally extends to the complexification ${\mathfrak g}_{\mathbf C}$ of ${\mathfrak g}_{\mathbf R}$, which gives rise to a holomorphic action of the complexification $G_{\mathbf C}^0$ of the neutral component $G_{\mathbf R}^0$ of the real Lie group $G_{\mathbf R}$. The latter action can easily be extended to a holomorphic action of $G_{\mathbf C}$ on the complex manifold $TM$. It is not hard to see that the action is transitive (for more details see e.g. Thm C of Szőke, R.: Automorphisms of certain Stein manifolds, Math. Z. 219 (1995), 357-385)