  Using an analogue of Makanin-Razborov diagrams, we give a description of the
solution set of systems of equations over an equationally Noetherian free
product of groups $G$. Equivalently, we give a parametrisation of the set
$Hom(H, G)$ of all homomorphisms from a finitely generated group $H$ to $G$.
Furthermore, we show that every algebraic set over $G$ can be decomposed as a
union of finitely many images of algebraic sets of NTQ systems.
  If the universal Horn theory of $G$ (the theory of quasi-identities) is
decidable, then our constructions are effective.
