I'd like to mention the work of Castaño-Bernard-Matessi-Solomon, who proved the existence of an anti-symplectic involution for symplectic manifolds carrying a Lagrangian torus fibration of a certain class. Such a class of Lagrangian fibration is constructed by gluing local models of Lagrangian fibrations to integral affine manifolds with singularities.
More precisely, let $(B,\mathscr{A},\Delta)$ be an integral affine manifold, whose affine structure $\mathscr{A}$ is singular along $\Delta\subset B$. In this case, we have a decomposition
$\Delta=\Delta_p\cup\Delta_g\cup\Delta_n$,
where the subscripts indicate the types of the singularity, namely positive, generic or negative. For simplicity, let's restricts to the case of a symplectic 6-manifold. After taking away the discriminant locus, $B\setminus\Delta$ carries the structure of an integral affine manifold, and we may form a Lagrangian torus bundle $T^\ast(B\setminus\Delta)/\Lambda^\ast$ in the obvious way, where $\Lambda\subset T(B\setminus\Delta)$ is a lattice bundle coming from the affine structure $\mathscr{A}$ on $B\setminus\Delta$ and $\Lambda^\ast$ denotes its dual. The aim is to glue local models of singular Lagrangian fibrations using fiber-preserving symplectomorphism to the torus bundle $T^\ast(B\setminus\Delta)/\Lambda^\ast$ so that we obtain a Lagrangian fibration $\pi:M\rightarrow B$ on the symplectic 6-manifold $M$ which serves as a compactification of the previous torus bundle.
For a small neighborhood $U_p\subset B$ so that $\Delta_p\subset U_p$, one considers the standard Lagrangian fibration $\pi_{std}:\mathbb{C}^3\setminus\{xyz=-1\}\rightarrow\mathbb{R}^3$ defined by
$(x,y,z)\rightarrow\left(\log|xyz+1|,\left(|x|^2-|y|^2\right)/2,\left(|x|^2-|z|^2\right)/2\right)$
Similarly one has local models of Lagrangian fibrations for $\Delta_g\subset U_g$ and $\Delta_n\subset U_n$. The outcome is a Lagrangian torus fibration $\pi:M\rightarrow B$ of preferred topology and singularities. In the paper of Castaño-Bernard-Matessi-Solomon, they call such Lagrangian fibrations belong to class $\mathscr{C}$.
For these Lagrangian fibrations, one can analyze their local models explicitly and glue local Lagrangian sections to a global one. One distinguished property of a Lagrangian fibration $\pi$ in $\mathscr{C}$ is that $\pi$ is not smooth near $\Delta_n$, so one has to impose some restrictions on the Lagrangian sections $\sigma$ of $\pi$ we work with, these restrictions specify a class $\mathfrak{C}$ of Lagrangian sections. With these preliminaries the main result of Castaño-Bernard-Matessi-Solomon can be stated as follows:
Theorem. Let $\pi:M\rightarrow B$ be a Lagrangian fibration of class $\mathscr{C}$. Given a Lagrangian section $\sigma$ of $\pi$ in class $\mathfrak{C}$, there is a unique antisymplectic involution $\phi_{\pi,\sigma}$ of $M$ such that $\phi_{\pi,\sigma}$ preserves the fibers of $\pi$ and fixes the Lagrangian section $\sigma$.
Notice that the class $\mathscr{C}$ is actually very general, it includes Lagrangian fibrations without singular fibers (cotangent bundles, tori), the so-called almost toric symplectic manifolds introduced by Leung-Symington (which plays an essential role in the recent deep work of Vianna in symplectic topology), and 6-dimensional symplectic Calabi-Yau manifolds homeomorphic to a quintic or Schoen's Calabi-Yau manifolds.
Speculations. From the point of view of mirror symmetry, the non-existence of an anti-symplectic involution roughly means the mirror of $M$ does not exist or at least carries a gerbe $\alpha\in H^2_{et}(M^\vee,\mathscr{O}^\ast)$. This is because the anti-symplectic involutions constructed by Castaño-Bernard-Matessi-Solomon (which are roughly reflections with respect to the Lagrangian section) is in some sense mirror to the functor $R\underline{Hom}(-,\mathscr{O}_{M^\vee})$ on the bounded derived category of coherent sheaves $D^b(M^\vee)$. It should be interesting to establish this speculation rigorously at least in some specific examples. But I don't know how to do this.