Summary: The non-abelian Hodge correspondence was established by K. Corlette [J. Differ. Geom. 28, No. 3, 361–382 (1988; Zbl 0676.58007)], S. K. Donaldson [Proc. Lond. Math. Soc. (3) 55, 127–131 (1987; Zbl 0634.53046)], N. J. Hitchin [Proc. Lond. Math. Soc. (3) 55, 59–126 (1987; Zbl 0634.53045)] and C. T. Simpson [J. Am. Math. Soc. 1, No. 4, 867–918 (1988; Zbl 0669.58008)], [Publ. Math., Inst. Hautes Étud. Sci. 75, 5–95 (1992; Zbl 0814.32003)]. It states that on a compact Kähler manifold \((X, \omega)\), there is a one-to-one correspondence between the moduli space of semi-simple flat complex vector bundles and the moduli space of poly-stable Higgs bundles with vanishing Chern numbers. In this paper, we extend this correspondence to the projectively flat bundles over some non-Kähler manifold cases. Firstly, we prove an existence theorem of Poisson metrics on simple projectively flat bundles over compact Hermitian manifolds. As its application, we obtain a vanishing theorem of characteristic classes of projectively flat bundles. Secondly, on compact Hermitian manifolds which satisfy Gauduchon and astheno-Kähler conditions, we combine the continuity method and the heat flow method to prove that every semi-stable Higgs bundle with \(\Delta \left( E,\bar{\partial}_E\right) \cdot \left[ \omega^{n-2}\right] = 0\) must be an extension of stable Higgs bundles. Using the above results, over some compact non-Kähler manifolds \((M,\omega)\), we establish an equivalence of categories between the category of semi-stable (poly-stable) Higgs bundles \((E,\bar{\partial}_E, \phi)\) with \(\Delta (E,\bar{\partial}_E) \cdot [\omega^{n-2}] = 0\) and the category of (semi-simple) projectively flat bundles \((E, D)\) with \(\sqrt{-1} F_D = \alpha \otimes\mathrm{Id}_E\) for some real \((1,1)\)-form \(\alpha\).