In general, the property of being real is not preserved by finite coverings, not even Galois ones.

For instance, take $X= \mathbb{P}^1$, which is a real curve with the anti-holomorphic involution $\sigma(z) = \bar{z}$. 

Now every elliptic curve $Y$ is a double cover of $\mathbb{P}^1$ branched in four points, but not all elliptic curves have a real structure.