  We study the ground-state phase diagram of the Ashkin-Teller random quantum
spin chain by means of a generalization of the strong-disorder renormalization
group. In addition to the conventional paramagnetic and ferromagnetic (Baxter)
phases, we find a partially ordered phase characterized by strong randomness
and infinite coupling between the colors. This unusual phase acts, at the same
time, as a Griffiths phase for two distinct quantum phase transitions both of
which are of infinite-randomness type. We also investigate the quantum
multi-critical point that separates the two-phase and three-phase regions; and
we discuss generalizations of our results to higher dimensions and other
systems.
