Principal bundles on compact complex manifolds with trivial tangent bundle

Let $G$ be a connected complex Lie group and $\mathrm{\Gamma }\subset G$ a cocompact lattice. Let $H$ be a complex Lie group. We prove that a holomorphic principal $H$-bundle $E_H$ over $G\mathrm{/}\mathrm{\Gamma }$ admits a holomorphic connection if and only if $E_H$ is invariant. If $G$ is simply connected, we show that a holomorphic principal $H$-bundle $E_H$ over $G\mathrm{/}\mathrm{\Gamma }$ admits a flat holomorphic connection if and only if $E_H$ is homogeneous.