  For a compact manifold with boundary $X$ we introduce the
  $n$-fold scattering stretched product $X^n_{\text{sc}}$ which is a compact
manifold with corners for each $n,$ coinciding with the previously known cases
for $n=2,3.$ It is constructed by iterated blow up of boundary faces and
boundary faces of multi-diagonals in $X^n.$ The resulting space is shown to map
smoothly, by a b-fibration, covering the usual projection, to the lower
stretched products. It is anticipated that this manifold with corners, or at
least its combinatorial structure, is a universal model for phenomena on
asymptotically flat manifolds in which particle clusters emerge at infinity. In
particular this is the case for magnetic monopoles on $\mathbb{R}^3$ in which
case these spaces are closely related to compactifications of the moduli spaces
with the boundary faces mapping to lower charge idealized moduli spaces.
