  We introduce a labelled version of the configuration spaces
$V(\mathbb{R}^{p,q})$ of vertically aligned clusters and prove a stable
splitting result for them. On the other hand, these spaces turn out to be
algebras over the little $(p+q)$-cubes operad $\mathscr{C}_{p+q}$. We will give
a geometric model for the $p$-fold bar construction, and additionally for the
$(p+q)$-fold bar construction in the case $q=1$.
