  The $N$-color quantum Ashkin-Teller spin chain is a prototypical model for
the study of strong-randomness phenomena at first-order and continuous quantum
phase transitions. In this paper, we first review the existing strong-disorder
renormalization group approaches to the random quantum Ashkin-Teller chain in
the weak-coupling as well as the strong-coupling regimes. We then introduce a
novel general variable transformation that unifies the treatment of the
strong-coupling regime. This allows us to determine the phase diagram for all
color numbers $N$, and the critical behavior for all $N \ne 4$. In the case of
two colors, $N=2$, a partially ordered product phase separates the paramagnetic
and ferromagnetic phases in the strong-coupling regime. This phase is absent
for all $N>2$, i.e., there is a direct phase boundary between the paramagnetic
and ferromagnetic phases. In agreement with the quantum version of the
Aizenman-Wehr theorem, all phase transitions are continuous, even if their
clean counterparts are of first order. We also discuss the various critical and
multicritical points. They are all of infinite-randomness type, but depending
on the coupling strength, they belong to different universality classes.
