Spin transfer from circularly polarized real photons to recoiling hyperons has been measured for the reactions γ⃗+p→K++Λ⃗ and γ⃗+p→K++Σ⃗0. The data were obtained using the CEBAF Large Acceptance Spectrometer (CLAS) detector at the Jefferson Lab for center-of-mass energies W between 1.6 and 2.53 GeV, and for -0.85θK+c.m.Cz, was found to be near unity for a wide range of energy and kaon production angles. The associated transverse polarization coefficient Cx is smaller than Cz by a roughly constant difference of unity. Most significantly, the total Λ polarization vector, including the induced polarization P, has magnitude consistent with unity at all measured energies and production angles when the beam is fully polarized. For the Σ0 this simple phenomenology does not hold. All existing hadrodynamic models are in poor agreement with these results.