It was shown recently that smooth and continuous ‘Matsubara’ phase-space loops follow a quantum-Boltzmann-conserving classical dynamics when decoupled from non-smooth distributions, which was suggested as the reason that many dynamical observables appear to involve a mixture of classical dynamics and quantum Boltz- mann statistics. Here we derive a mean-field version of this ‘Matsubara dynamics’ which sufficiently mitigates its serious phase problem to permit numerical tests on a two-dimensional ‘champagne-bottle’ model of a rotating OH bond. The Matsubara- dynamics rovibrational spectra are found to converge towards close agreement with the exact quantum results at all temperatures tested (200–800 K), the only significant discrepancies being a temperature-independent 22 cm−1 blue-shift in the position of the vibrational peak, and a slight broadening in its lineshape. These results are compared with centroid molecular dynamics (CMD) to assess the importance of non- centroid fluctuations. Above 250 K, only the lowest-frequency non-centroid modes are needed to correct small CMD red-shifts in the vibrational peak; below 250 K, more non-centroid modes are needed to correct large CMD red-shifts and broaden- ing. The transition between these ‘shallow curvature’ and ‘deep curvature’ regimes happens when imaginary-time Feynman paths become able to lower their actions by cutting through the curved potential surface, giving rise to artificial instantons in CMD.