Let $M_d^{(k)}(n)$ be the manifold of $n$-tuples $(x_1,\ldots,x_n)\in(\mathbb{R}^d)^n$ having non-$k$-equal coordinates. We show that, for $d\geq 2$, $M_{d}^{(3)}(n)$ is rationally formal if and only if $n\leq6$. This stands in sharp contrast with the fact that all classical configuration spaces $M_{d}^{(2)}(n)=(\mathbb{R}^d,n)$ are rationally formal, just as are all complements of arrangements of arbitrary complex subspaces with geometric lattice of intersections. The rational non-formality of $M_{d}^{(3)}(n)$ for $n>6$ is established via detection of non-trivial triple Massey products, which are assessed geometrically through Poincaré duality.