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

More precisely, $Y$ is real if and only if the affine equation of the double cover $Y \to \mathbb{P}^1$ is of the form

$w^2=(z-a)(z-\bar{a})(z-b)(z-\bar{b}) \quad a,b \in \mathbb{C}$.

In this case there are exactly two liftings of $\sigma$ to $Y$, namely

$(z,w) \to (\bar{z}, \bar{w}) \quad $ and $ \quad (z,w) \to (\bar{z}, -\bar{w})$.

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

More precisely, $\sigma$ can be lifted to $Y$ if and only if the double cover $Y \to \mathbb{P}^1$ admits an affine equation of the form

$w^2=(z-a)(z-\bar{a})(z-b)(z-\bar{b}) \quad a,b \in \mathbb{C}$.

In this case there are exactly two liftings, namely

$(z,w) \to (\bar{z}, \bar{w}) \quad $ and $ \quad (z,w) \to (\bar{z}, -\bar{w})$.

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.

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

More precisely, $Y$ is real if and only if the affine equation of the double cover $Y \to \mathbb{P}^1$ is of the form

$w^2=(z-a)(z-\bar{a})(z-b)(z-\bar{b}) \quad a,b \in \mathbb{C}$.

In this case there are exactly two liftings of $\sigma$ to $Y$, namely

$(z,w) \to (\bar{z}, \bar{w}) \quad $ and $ \quad (z,w) \to (\bar{z}, -\bar{w})$.