  Dipolar-octupolar pyrochlore magnets in a strong external magnet field
applied in the [110] direction are known to form a `chain' state, with
subextensive degeneracy. Magnetic moments are correlated along one-dimensional
chains carrying effective Ising degrees of freedom which are noninteracting on
the mean-field level. Here, we investigate this phenomenon in detail, including
the effects of quantum fluctuations. We identify two distinct types of chain
phases, both featuring distinct subextensive, classical ground state
degeneracy. Focussing on one of the two kinds, we discuss lifting of the
classical degeneracy by quantum fluctuations. We map out the ground-state phase
diagram as a function of the exchange couplings, using linear spin wave theory
and real-space perturbation theory. We find a hierarchy of energy scales in the
ground state selection, with the effective dimensionality of the system varying
in an intricate way as the hierarchy is descended. We derive an effective
two-dimensional anisotropic triangular lattice Ising model with only three free
parameters which accounts for the observed behavior. Connecting our results to
experiment, they are consistent with the observation of a disordered chain
state in Nd$_2$Zr$_2$O$_7$. We also show that the presence of two distinct
types of chain phases has consequences for the field-induced breakdown of the
apparent $U(1)$ octupolar quantum liquid phase recently observed in
Ce$_2$Sn$_2$O$_7$.
