  We determine the ground states and excitation spectra of the paradigmatic
four-flavour Heisenberg model with nearest- and next-nearest-neighbor exchange
couplings on the triangular lattice in a field controlling the population
imbalance of flavor pairs. Such a system arises in the strongly correlated
limit of moir\'e bilayers of transition metal dichalcogenides in an electric
displacement field or in-plane magnetic field, and can be simulated via
ultracold alkaline-earth atoms. We argue that the field tunes between effective
SU(4) and SU(2) symmetries in the balanced and fully polarised limits and
employ a combination of mean-field calculations, flavour-wave theory, and exact
diagonalisation to analyse the intermediate, imbalanced regime. We find
different symmetry-broken phases with simultaneous spin and excitonic order
depending on the field and next-nearest-neighbor coupling. Furthermore, we
demonstrate that there is a strongly fluctuating regime without long-range
order that connects candidate spin liquids of the SU(2) and SU(4) limit. The
strong fluctuations are facilitated by an extensive classical degeneracy of the
model, and we argue that they are also responsible for a strong polarisability
at 1/3 polarisation that survives from the mean-field level to the exact
spectrum.
