Is there a Kähler manifold with no anti-holomophic involution? That is, is there a Kähler manifold $X$ on which there is no map
$$
\tau:X\to X
$$
such that
$$
d\tau\circ I=-I\circ d\tau
$$
and
$$
\tau\circ \tau=\mathrm{Id}_X?
$$
 A: There even exists a (compact) Kähler manifold with no anti-holomorphic self-diffeomorphism at all. Namely a suitable elliptic curve.
Indeed, consider $\mathbf{C}/\Lambda$, for some lattice $\Lambda$. Let $f$ be an anti-holomorphic self-diffeomorphism. Up to compose with a translation, we can suppose that $f(0)=0$. Let $F$ be the unique lift $\mathbf{C}\to\mathbf{C}$ mapping $0$ to $0$. Then $F$ is a self-covering, and hence is a diffeomorphism, and commutes with $\Lambda$-translations. Write $G(z)=\overline{F(z)}$. Then $G$ is a holomorphic self-diffeomorphism of $\mathbf{C}$ and fixes $0$, hence $G(z)=\bar{q}z$ for some $q\in\mathbf{C}^*$. Since $G$ maps $\Lambda$ to $\overline{\Lambda}$ which has the same covolume, we have $|q|=1$. We have $F(z)=q\bar{z}$ for all $z$, and $F(\Lambda)=\Lambda$. So $\Lambda=q\overline{\Lambda}$. That is, $\Lambda$ is preserved by some reflection $z\mapsto q\bar{z}$.
Next is it enough to see that "most" lattices are not preserved by any reflection. Indeed, we can choose coordinates so that $\Lambda$ has the basis $(1,z)$ with $z$ belonging to the usual "strip" ($|z|\ge 1$, $\frac12<\mathrm{Re}(z)\le \frac12$, $\mathrm{Im}(z)>0$). Then such a lattice is preserved by a reflection if and only if $\mathrm{Re}(z)\in\{0,1/2\}$. (The latter point was already observed in the previous answer using another language, but the easy argument could easily be explained.)
(Indeed we see in these cases that there exists a anti-holomorphic self-diffeomorphism if and only if there is one that is in addition involutive, and furthermore preserves the standard flat Kähler structure.)
A: The moduli space $\mathcal{M}_1^\mathbb{R}$ of real algebraic curves of genus $1$ equals the real part of the moduli space $\mathcal{M}_1=\mathbb{C}$. In particular, the general elliptic curve has no anti-holomorphic involution, as suggested by P. Achinger in his comment.
Furthermore, similar statements hold also for real curves of higher genera. See
Seppälä, Mika, Real algebraic curves in the moduli space of complex curves, Compos. Math. 74, No. 3, 259-283 (1990). ZBL0725.14019.
