  We study the low-energy effective theories of photons and gravitons, matching
them at one loop to a UV completion with a massive spinning matter particle.
The matter is allowed to have non-minimal electromagnetic and gravitational
interactions. We construct the one-loop 4-photon and 4-graviton helicity
amplitudes with matter in the loop carrying anomalous multipole moments, and we
read off the Wilson coefficients in the effective theory below the matter
particle mass. Much as in the case of minimal coupling, the Wilson coefficients
beyond the leading order turn out to be confined to small islands in the much
larger theory space allowed by existing positivity constraints.
