The consistency of the EFT of two interacting spin-2 fields is checked by
applying forward limit positivity bounds on the scattering amplitudes to
exclude the region of parameter space devoid of a standard UV completion. We
focus on two classes of theories that have the highest possible EFT cutoff,
namely those theories modelled on ghost-free interacting theories of a single
massive spin-2 field. We find that the very existence of interactions between
the spin-2 fields implies more stringent bounds on all the parameters of the
EFT, even on the spin-2 self-interactions. This arises for two reasons. First,
with every new field included in the low-energy EFT, comes the `knowledge' of
an extra pole to be subtracted, hence strengthening the positivity bounds.
Second, while adding new fields increases the number of free parameters from
the new interactions, this is rapidly overcome by the increased number of
positivity bounds for different possible scattering processes. We also discuss
how positivity bounds appear to favour relations between operators that
effectively raise the cutoff of the EFT.