Riesz homomorphisms on vector lattices have been generalized to Riesz* homomorphisms on ordered vector spaces by van Haandel using a condition on sets of finitely many elements. Van Haandel attempted to prove that it suffices to take sets of two elements. We show that this is not true, in general. The description by two elements motivates to introduce mild Riesz* homomorphisms. We investigate their properties on order unit spaces, where the geometry of the dual cone plays a crucial role. Hereby, we mostly focus on the finite-dimensional case.