  We examine the late-time (t > 200 days after peak brightness) spectra of Type
Iax supernovae (SNe Iax), a low-luminosity, low-energy class of thermonuclear
stellar explosions observationally similar to, but distinct from, Type Ia
supernovae. We present new spectra of SN 2014dt, resulting in the most complete
published late-time spectral sequence of a SN Iax. At late times, SNe Iax have
generally similar spectra, all with a similar continuum shape and strong
forbidden-line emission. However, there is also significant diversity where
some late-time SN Iax spectra display narrow P-Cygni features and a continuum
indicative of a photosphere in addition to strong narrow forbidden lines, while
others have no obvious P-Cygni features, strong broad forbidden lines, and weak
narrow forbidden lines. Finally, some SNe Iax have spectra intermediate to
these two varieties with weak P-Cygni features and broad/narrow forbidden lines
of similar strength. We find that SNe Iax with strong broad forbidden lines
also tend to be more luminous and have higher-velocity ejecta at peak
brightness. We estimate blackbody and kinematic radii of the late-time
photosphere, finding the latter an order of magnitude larger than the former.
We propose a two-component model that solves this discrepancy and explains the
diversity of the late-time spectra of SNe Iax. In this model, the broad
forbidden lines originate from the SN ejecta, while the photosphere, P-Cygni
lines, and narrow forbidden lines originate from a wind launched from the
remnant of the progenitor white dwarf and is driven by the radioactive decay of
newly synthesized material left in the remnant. The relative strength of the
two components accounts for the diversity of late-time SN Iax spectra. This
model also solves the puzzle of a long-lived photosphere and slow late-time
decline of SNe Iax. (Abridged)
