  Recent experiments have examined the impact of a magnetic field on
ferroquadrupolar orders in the intermetallic Kondo material PrTi$_2$Al$_{20}$.
Motivated by this, we use extensive Monte Carlo simulations to study a diamond
lattice XY model of non-Kramers pseudospin-$1/2$ Pr$^{3+}$ moments which
crucially incorporates three-spin interactions. This model supports a thermal
$Z_3$ Potts ordering transition upon cooling from the paramagnetic phase into
the ferroquadrupolar phase. An applied magnetic field along the [110] direction
leads to a thermal Ising transition out of the quadrupolar ordered phase. A
magnetic field along the [001] direction leads to only thermal crossovers, but
supports a spinodal transition out of metastable domains which could be
strongly pinned by coupling to elastic lattice deformations. We propose noise
measurements as a potential probe to "hear" the spinodal transition. Our work
highlights the importance of multispin interactions in Kondo materials near the
small-to-large Fermi surface transition.
