A manifold is called geometrically formal when all wedge products of harmonic forms are harmonic.
As it says here: https://arxiv.org/pdf/math/0004009.pdf: On a general Riemannian manifold, wedge products of harmonic forms are not usually harmonic. But there are some examples where this does happen, like compact globally symmetric spaces. For these the harmonic forms coincide with the invariant ones, and the latter are clearly closed under products.
I am not so used to symmetric spaces, my question is: Is a complex tori $\mathbb{C}^n/\Lambda$ for some lattice $\Lambda$ globally symmetric space and hence formal?
These properties don't require a complex structure, so there's no use in distinguishing complex tori from real tori here.
Tori $\mathbb{R}^n/\Gamma$ (equipped with the corresponding flat metric) are not symmetric spaces for $n>1$, since their isotropy groups are finite. They are formal, however, since a form on such a torus is harmonic iff it has constant coefficients.