Energy transfer in noncovalently bound porphyrin/carbon nanotube compounds is investigated at the single-nanocompound scale. Excitation spectroscopy of the luminescence of the nanotube shows two resonances arising from intrinsic excitation of the nanotube and from energy transfer from the porphyrin. Polarization diagrams show that both resonances are highly anisotropic, with a preferred direction along the tube axis. The energy transfer is thus strongly anisotropic despite the almost isotropic absorption of porphyrins. We account for this result by local field effects induced by the large optical polarizability of nanotubes. We show that the local field correction extends over several nanometers outside the nanotubes and drives the overall optical response of functionalized nanotubes.