© Foundation Compositio Mathematica 2014. A flag domain D = G/V for G a simple real non-compact group G with compact Cartan subgroup is non-classical if it does not fiber holomorphically or anti-holomorphically over a Hermitian symmetric space. We prove that for Γ an infinite, finitely generated discrete subgroup of G, the analytic space Γ / D does not have an algebraic structure. We also give another proof of the theorem of Huckleberry that any two points in a non-classical domain D can be joined by a finite chain of compact subvarieties of D.