An inverse Clifford Semigroup is a semilattice of groups. Conditions are given for constructing a compact semitopological (separately continuous multiplication) inverse Clifford semigroup on a compact Hausdorff semilattice. The conditions are necessary and sufficient for decomposing a compact inverse Clifford semigroup containing a dense subgroup and locally compact maximal groups into its semilattice of groups. A catalogue of examples is given to demonstrate the construction while exhibiting various pathologies.