Perhaps my comment was a bit too cryptic, so let me expand it slighty. If $X\subset \mathbb{P}^n_\mathbb{C}=\mathbb{P}$ is defined by real polynomials, then conjugation $\tau:[x_0,\ldots, x_n]\mapsto [\bar x_0,\ldots, \bar x_n]$ induces an action on $X$. The story for a general coherent sheaf $F$ is similar. It can always be given as the cokernel of a matrix $\oplus \mathcal{O}_{\mathbb{P}}(a_i)\to \oplus \mathcal{O}_{\mathbb{P}}(b_j)$. In order to get a natural action of $\tau$ on $F$, which is an isomorphism $\tau^*F\cong F$ with "square" equal the identity, it would be enough to assume that some presentation matrix is given by real polynomials. To put it more canonically, the pair $(X,F)$ should be obtained by base change for a pair defined over $Spec\mathbb{R}$.

Perhaps my comment was a bit too cryptic, so let me expand it slighty. If $X\subset \mathbb{P}^n_\mathbb{C}=\mathbb{P}$ is defined by real polynomials, then conjugation $\tau:[x_0,\ldots, x_n]\mapsto [\bar x_0,\ldots, \bar x_n]$ induces an action on $X$. The story for a general coherent sheaf $F$ is similar. It can always be given as the cokernel of a matrix $\oplus \mathcal{O}_{\mathbb{P}}(a_i)\to \oplus \mathcal{O}_{\mathbb{P}}(b_j)$. In order to get a natural action of $\tau$ on $F$, which is an isomorphism $\tau^*F\cong F$ with "square" equaling the identity, it would be enough to assume that some presentation matrix is given by real polynomials. To put it more canonically, the pair $(X,F)$ should be obtained by base change from a pair defined over $Spec\mathbb{R}$.