Let us first consider the singular cohomology group $H^i(X(\mathbb{C}),\mathbb{C})$. There are three ways to let the group $G:=\mathrm{Gal}(\mathbb{C}/\mathbb{R})\simeq\mathbb{Z}/2$ act on it : by letting complex conjugation act on the coefficients $\mathbb{C}$, or on the space $X(\mathbb{C})$, or on both. Denote by $F_{\mathrm{B}}$, $F_{\infty}$ and $F_{\mathrm{dR}}$ these three involutions of $H^i(X(\mathbb{C}),\mathbb{C})$. They commute and $F_{\mathrm{dR}}=F_{\infty}\circ F_{\mathrm{B}}$. The involution $F_{\infty}$ is $\mathbb{C}$-linear, and $F_B$ and $F_{\mathrm{dR}}$ are $\mathbb{C}$-antilinear. The involution $F_{\mathrm{B}}$ exists for all complex varieties (and is the antiholomorphic involution associated with the real structure $H^i(X(\mathbb{C}),\mathbb{R})$ of $H^i(X(\mathbb{C}),\mathbb{C})$), but defining $F_{\infty}$ and $F_{\mathrm{dR}}$ requires $X$ to be defined over $\mathbb{R}$. The involution $F_{\mathrm{dR}}$ preserves the Hodge decomposition (as the next paragraph shows), but $F_{\mathrm{B}}$ and hence $F_{\infty}$ reverse it.

Now consider the Grothendieck comparison theorem $H^i(X(\mathbb{C}),\mathbb{C})=H^i_{\mathrm{dR}}(X_{\mathbb{C}}/\mathbb{C})$ between singular and algebraic de Rham cohomology. Inspecting the proof of this theorem shows that the antiholomorphic involution of this space associated with the real structure $H^i_{\mathrm{dR}}(X/\mathbb{R})$ of $H^i_{\mathrm{dR}}(X_{\mathbb{C}}/\mathbb{C})$ is $F_{\mathrm{dR}}$. Consequently, $F_{\mathrm{B}}$ and $F_{\infty}$ descend to the same $\mathbb{R}$-linear involution on $H^i_{\mathrm{dR}}(X/\mathbb{R})$ (let us still call it $F_{\infty}$). It is with respect to this involution, which is very much nontrivial and difficult to compute on the de Rham side, that $F^pH^i_{\mathrm{dR}}(X/\mathbb{R})\cap F_{\infty}(F^qH^i_{\mathrm{dR}}(X/\mathbb{R}))=0$ when $p+q>i$.