It is shown in Gudder and Schelp (1970) that partial Baer *-semigroups coordinatize orthomodular partially ordered sets (orthomodular posets). This means for P an orthomodular poset there exists a partial Baer *-semigroup whose closed projections are order isomorphic to P preserving ortho-complementation. This coordinatization theorem generalizes Foulis (1960) in which orthomodular lattices are coordinatized by Baer *-semigroups. In particular Foulis (unpublished) shows that any complete atomic Boolean lattice is coordinatized by a Bear *-semigroup of relations. Since Greechie (1968), (1971) shows that a whole class of orthomodular posets can be formed by “pasting” together Boolean lattices, it is natural to consider the following problem. Let y be a family of Baer *-semigroups of relations which coordinatize the family B of complete atomic lattices. Is it possible to construct a partial *-semigroup of relations R which contains each member of Y such that when P is an orthomodular poset obtained by a “Greechie pasting” of members of 38 then 91 coordinatizes R This question is considered in the sequel and answered affirmatively for a certain subclass of “Greechie pasted” orthomodular posets. In addition the construction of 8)t nicely fulfills another objective in that it provides us with “nontrivial” coordinate partial Baer *-semigroups for a whole family of well known orthomodular posets. This is particularly significant since the only other known coordinate partial Baer *-semigroups, for those posets in this family which are not lattices, are the “minimal” ones given in Gudderand and Schelp (1970).