What follows could be formulated for more general extensions than $\mathbb{R}\rightarrow\mathbb{C}$ but I'll stick to this particular case for now. Further, I am somewhat new to this language and I'm suspicious I'm making a number of silly mistakes, so be warned!
Let $X$ be a complex variety, definable over $\mathbb{R}$, so that $X$ is equipped with an isomorphism $X\rightarrow X^{\sigma}$, where $\sigma$ denotes complex conjugation. I believe this allows us to ask the question of when a quasi coherent sheaf on $X$ has a $\mathbb{R}$ structure. Further, it should be the case that any "natural" sheaf is canonically endowed with such a structure.
Let $\mathcal{M}$ be the moduli stack of stable elliptic curves over $\mathbb{C} $, so a DM-stack, but the discussion above should still apply. I want to know when a line bundle on $\mathcal{M}$ admits a real structure. Consider for example the Hodge line bundle $\omega$, whose fiber over a point of $\mathcal{M} $ corresponding to an elliptic curve $E$ is $H^{0}(E,\Omega)$. Is this bundle endowed with a real structure? It doesn't seem to me that it should be, as $H^{1,0}$ is rather identified naturally with the complex conjugate to $H^{0,1}$. Nonetheless it's tensor square should be, as it's the canonical bundle on $\mathcal{M}$...