It seems to me that you could be interested in the following (I haven't checked the paper in detail, but I think theorems of this "style" could be helpful for you):

 - Dieter Kotschick,  [Orientations and geometrisations of compact complex surfaces][1]  (Bull. London Math. Soc. 29 (1997), no. 2, 145–149. [Zbl 0896.32014][2])

> **Theorem** Let $X$ be a compact complex surface admitting a complex structure for $\bar{X}$. Then $X$ (and $\bar{X}$) satisfies one of the following:
> 1. $X$ is geometrically ruled, or
> 2. the Chern numbers $c_1^2$ and $c_2$ of $X$ vanish, or
> 3. $X$ is uniformised by the polydisk.
>
>In particular, the signature of $X$ vanishes.


Other material that could be helpful is:

 - Dieter Kotschick, [Orientation-reversing homeomorphisms in surface
geography][3] (Math. Ann. 292 (1992), no. 2, 375–381. [Zbl 0753.14034][4])
 - Arnaud Beauville, [Surfaces complexes et orientation][5] (Astérisque 126 (1985), 41–43. [Zbl 0574.14032][6])


  [1]: https://doi.org/10.1112/S0024609396002287
  [2]: https://zbmath.org/?q=an:0896.32014
  [3]: https://doi.org/10.1007/BF01444627
  [4]: https://zbmath.org/?q=an:0753.14034
  [5]: http://www.numdam.org/item/?id=AST_1985__126__41_0
  [6]: https://zbmath.org/?q=an:0574.14032