The holonomy group of a Riemannian manifold always arises in geometry through a particular representation, not as an abstract group. One can therefore ask whether there exist pairs of (compact, locally irreducible) manifolds with holonomy groups which are isomorphic, yet distinct, because the holonomy representations are not equivalent. A theorem of Besse asserts that this is not possible in the simply connected case; however, it is possible for certain nonsimply connected manifolds. Here we identify all of these manifolds (up to space form problems) in the case where the Ricci curvature is not negative. This allows us to solve the holonomy classification problem for all compact, locally irreducible Riemannian manifolds of positive Ricci curvature.