  We study a one-dimensional ladder of two coupled XXZ spin chains and identify
several distinct gapless symmetry-enriched critical phases. These have the same
unbroken symmetries and long-wavelength description, but cannot be connected
without encountering either a phase transition or other intermediate phases.
Using bosonizaion, we analyze the nature of their distinction by determining
how microscopic symmetries are manifested in the long-wavelength fields, the
behavior of charged local and nonlocal operators, and identify the universality
class of all direct continuous phase transitions between them. One of these
phases is a gapless topological phase with protected edge modes. We
characterize its precise nature and place it within the broader classification.
We also find the occurrence of `multiversality' in the phase diagram, wherein
two fixed phases are separated by continuous transitions with different
universality classes in different parameter regimes. We determine the phase
diagram and all its aspects, as well as verify our predictions numerically
using density matrix renormalization group and a mapping onto an effective
spin-1 model.
