  We use the recently proposed generalised on-shell representation for
scattering amplitudes and a consistency test to explore the space of tree-level
consistent couplings in four-dimensional Minkowski spacetime. The extension of
the constructible notion implied by the generalised on-shell representation,
i.e. the possibility to reconstruct at tree level all the scattering amplitudes
from the three-particle ones, together with the imposition of the consistency
conditions at four-particle level, allow to rediscover all the known theories
and their algebra structure, if any. Interestingly, this analysis seems to
leave room for high-spin couplings, provided that at least the requirement of
locality is weakened. We do not claim to have found tree-level consistent
high-spin theories, but rather that our methods show signatures of them and
very likely, with a suitable modification, they can be a good framework to
perform a systematic search.
