  We discuss the effects of exponential fragmentation of the Hilbert space on
phase transitions in the context of coupled ferromagnetic Ising models in
arbitrary dimension with special emphasis on the one dimensional case. We show
that the dynamics generated by quantum fluctuations is bounded within spatial
partitions of the system and weak mixing of these partitions caused by global
transverse fields leads to a zero temperature phase with ordering in the local
product of both Ising copies but no long range order in either species. This
leads to a natural connection with the Ashkin-Teller universality class for
general lattices. We confirm this for the periodic chain using quantum Monte
Carlo simulations. We also point out that our treatment provides an explanation
for pseudo-first order behavior seen in the Binder cumulants of the classical
frustrated $J_1-J_2$ Ising model and the $q=4$ Potts model in 2D.
