  We propose a novel construction of finite hypergraphs and relational
structures that is based on reduced products with Cayley graphs of groupoids.
To this end we construct groupoids whose Cayley graphs have large girth not
just in the usual sense, but with respect to a discounted distance measure that
contracts arbitrarily long sequences of edges within the same sub-groupoid
(coset) and only counts transitions between cosets. Reduced products with such
groupoids are sufficiently generic to be applicable to various constructions
that are specified in terms of local glueing operations and require global
finite closure. We here examine hypergraph coverings and extension tasks that
lift local symmetries to global automorphisms.
