Let $M$ be a complex manifold. Consider a connection $\nabla$ on the holomorphic tangent bundle $T^{1,0}M$. The *torsion* of $\nabla$ is defined as the torsion of the induced connection $D$ on the real tangent bundle, $$T_\nabla(\alpha,\beta) = T_D(\alpha,\beta) := D_\alpha \beta - D_\beta \alpha - [\alpha,\beta]$$ for smooth vector fields $\alpha$ and $\beta$. The connection $\nabla$ is *torsion-free* if $T_\nabla = 0$.

Assume now that $\nabla$ is torsion-free. It is well-known that if $\nabla$ is also hermitian, i.e., compatible with a hermitian metric $h$ on $T^{1,0}M$, then it is the Chern connection of the metric, i.e., $\nabla$ is a $(1,0)$-connection in the sense that $\nabla^{0,1}=\bar\partial$, and $h$ provides a Kähler metric on $M$, see e.g., Huybrechts, Complex Geometry, Proposition 4.A.7.

I am interested in whether one can find torsion-free $(1,0)$-connections on $T^{1,0} M$ also on non Kähler manifolds? Such connections would thus necessarily not be hermitian by the above result. Or is there some obstruction?