We study coupled unitary Virasoro minimal models in the large rank (m→∞) limit. In large m perturbation theory, we find two nontrivial IR fixed points which exhibit irrational coefficients in several anomalous dimensions and the central charge. For N>4 copies, we show that the IR theory breaks all possible currents that would otherwise enhance the Virasoro algebra, up to spin 10. This provides strong evidence that the IR fixed points are examples of compact, unitary, irrational conformal field theories with the minimal amount of chiral symmetry. We also analyze anomalous dimension matrices for a family of degenerate operators with increasing spin. These display further evidence of irrationality and begin to reveal the form of the leading quantum Regge trajectory.