  Understanding and predicting lattice dynamics in strongly anharmonic crystals
is one of the long-standing challenges in condensed matter physics. Here we
propose a first-principles method that gives accurate quasiparticle (QP) peaks
of the phonon spectrum with strong anharmonic broadening. On top of the
conventional first-order self-consistent phonon (SC1) dynamical matrix, the
proposed method incorporates frequency renormalization effects by the bubble
self-energy within the QP approximation. We apply the developed methodology to
the strongly anharmonic $\alpha$-CsPbBr$_3$ that displays phonon instability
within the harmonic approximation in the whole Brillouin zone. While the SC1
theory significantly underestimates the cubic-to-tetragonal phase transition
temperature (\tc) by more than 50\%, we show that our approach yields \tc =
404--423~K, in excellent agreement with the experimental value of 403~K. We
also demonstrate that an accurate determination of QP peaks is paramount for
quantitative prediction and elucidation of phonon linewidth..
