  We establish faithful tropicalisation for point configurations on algebraic
tori. Building on ideas from enumerative geometry, we introduce tropical
scaffolds and use them to construct a system of modular fan structures on the
tropical configuration spaces. The corresponding toric varieties provide
modular compactifications of the algebraic configuration spaces, with boundary
parametrising transverse configurations on tropical expansions. The rubber
torus, used to identify equivalent configurations, plays a key role. As an
application, we obtain a modular interpretation for the bipermutahedral
variety.
