We study the structural phase transition associated with the highest
superconducting critical temperature measured in high-pressure sulfur hydride.
A quantitative description of its pressure dependence has been elusive for any
\emph{ab initio} theory attempted so far, raising questions on the actual
mechanism driving the transition. Here, we reproduce the critical pressure of
the hydrogen bond symmetrization in the Im3Λm structure, in agreement
with experimental data, by combining quantum Monte Carlo simulations for
electrons with path integral molecular dynamics for quantum nuclei. For
comparison, we also apply the self-consistent harmonic approximation, which
underestimates the critical pressure by about 40 GPa even when the most
accurate potential energy surface is used, pinpointing the importance of an
exact treatment of nuclear quantum effects. They indeed play a major role in a
significant reduction (β 100 GPa) of the classical transition pressure
and in a large isotope shift (β 25 GPa) upon hydrogen-to-deuterium
substitution