Your question is correspond to weak Kahler-Einstein metric(I know for Kahler case). Let me explain in more details
The Hermitian metric $H$ along the fibers of the holomorphic vector bundle $E$ over a K"ahler manifold $(M,\omega)$ is weak Hermitian-Einstein if $$\sqrt[]{-1}\Lambda_\omega F_H = \phi I$$ at every point of $M$, where $\gamma$ is a function on $M$ and $I$ is the identity endomorphism of $E$ and $F_H$ is the Chern curvature form of $(E, h)$. Here $\Lambda_\omega$ is the contraction with $\omega$ which means $$F_H\wedge\omega^{n-1}=(n-1)!\Lambda_\omega F_H\frac{\omega^n}{n!}$$
If $\Phi$ is constant $\lambda$ then we call it Hermitian-Einstein metric. As additional remark of the sign of $\lambda$,
If the metric $h$ admit hermitian-Einstein metric $Λ_ωF_h=λI $, and if $λ<0$ there is no holomorphic section, and if $λ=0$ then all holomorphic sections are parallel. Since $$\Delta |s|^2=|\nabla s|^2-<(\Lambda_\omega F_h)s,s>\geq 0$$
where $s$ is a section and hence $|s|$ is bounded. But maximum principle for subharmonic functions, so $|∇s|^2=λ|s|^2$. Hence if $λ<0$ there is no holomorphic section, and if $λ=0$ then all holomorphic sections are parallel.
Now take $E=TM$, then we can define weak Kahler-Einstein metric as $Ric(\omega)=\phi \omega$ and hence $c_1(TM)=[Ric(\omega)]=[\phi \omega]$
When $\phi$ is globally constant, then we call it Kahler-Einstein metric and when it is positive we call $M$ is Fano and when it is negative we call $M$ is of general type and it is zero $M$ is called Calabi-Yau variety
From Kobayashi book (complex vector bundle ) we have the following theorem for more general setting of your question
Theorem: An Hermitian vector bundle $(E, h)$ over a Riemann surface $(M, g)$ satisfies the weak Einstein condition if and only if it is projectively flat.
Note that in general one direction always works
Theorem: Let $(M, g)$ is a Kahler manifold, Let $(E, h)$ be a projectively flat vector bundle over $M$. Then, for any Hermitian metric $g$ on $M$,$(E, h)$ satisfies the weak Einstein condition.
We have also the following theorem which tell us that any weak Hermitian-Einstein metric after conformal change is Hermitian-Einstein metric
Theorem: If an Hermitian vector bundle $(E, h)$ over a compact Kahler manifold $(M, g)$ satisfies the weak Einstein condition with factor $φ$, there is a conformal change $h −→ h′ = ah$ such that $(E, h′ )$ satisfies the Einstein condition with a constant factor $c$. Such a conformal change is unique up to a homothety. Now you can take $E=TM$ and this tells us that any weak Kahler-Einstein there is a conformal change $h −→ h′ = ah$ such that $(TM, h′ )$ is Kahler-Einstein metric