Let $K\subset L$ be a finite galois extension of fields (the case I have in mind is $K=\mathbb{R},L=\mathbb{C}$). Given a scheme $X$ over $K$ by pulling back to $L$ we get a scheme $Y=X\times _K L$ over $L$ with a $K$ - action of $G=\mathrm{Gal}(L/K)$. It is easy to see that this action satisfies the following: There exists an affine open cover $X=\bigcup_{i\in I}U_i $ such that any $\sigma \in G$ takes each $U_i$ to itself. Now let $Y$ be an arbitrary scheme over $L$ with a $K$ - action of $G$ that satisfies the above property (for all of this you can assume whatever finiteness and niceness conditions you want on everything, the main case I have in mind is that of finite type separated schemes over the base fields, not necessarily reduced or affine and if possible not necessarily quasi projective). My questions are the following:

1) What other necessary conditions on the action of $G$ on $Y$ (other then the one I stated) must I assume to have a chance of $Y$ coming from a scheme $X$ over $K$?

2) Are there any simple sufficient conditions for $Y$ to come from such an $X$?

3)What are important examples of schemes $Y$ over $\mathbb{C}$ with a $\mathrm{Gal}(\mathbb{C}/\mathbb{R})$ action *not* coming from a scheme over $\mathbb{R}$ I should keep in mind? If possible, I want all actions to be anti-holomorphic i.e. inducing anti-holomorphic homomorphisms on the stalks of the analytification. Interesting examples for analytic spaces which don't come from schemes will also be very nice.

Edit: Assume throughout that the structural map $Y\rightarrow \mathrm{Spec(L)}$ is $G$-equivariant to avoid redundancies, thanks to Julian Rosen for pointing this out.