  We employ THz-range temperature-dependent Raman spectroscopy and
first-principles lattice-dynamical calculations to show that the undoped sodium
ion conductors Na$_3$PS$_4$ and isostructural Na$_3$PSe$_4$ both exhibit
anharmonic lattice dynamics. The anharmonic effects in the compounds involve
coupled host lattice -- Na$^+$ ion dynamics that drive the tetragonal-to-cubic
phase transition in both cases, but with a qualitative difference in the
anharmonic character of the transition. Na$_3$PSe$_4$ shows almost purely
displacive character with the soft modes disappearing in the cubic phase as the
change of symmetry shifts these modes to the Raman-inactive Brillouin zone
boundary. Na$_3$PS$_4$ instead shows order-disorder character in the cubic
phase, with the soft modes persisting through the phase transition and
remaining active in Raman in the cubic phase, violating Raman selection rules
for that phase. Our findings highlight the important role of coupled host
lattice -- mobile ion dynamics in vibrational instabilities that are coincident
with the exceptional conductivity in these Na$^+$ ion conductors.
