  Let $M^{(k)}_{d}(n)$ 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 $d\geq2$, $M^{(3)}_{d}(n)$ is rationally formal if and only if $n\leq
6$. This stands in sharp contrast with the fact that all classical
configuration spaces $M^{(2)}_d(n)=\text{Conf}(\hspace{.2mm}\mathbb{R}^d,n)$
are rationally formal, just as are all complements of arrangements of arbitrary
complex subspaces with geometric lattice of intersections. The rational non
formality of $M^{(3)}_{d}(n)$ for $n>6$ is established via detection of
non-trivial triple Massey products assessed through Poincar\'e duality.
