The φ-Λ(1520) interference effect in the γp→K+K-p reaction has been measured for the first time in the energy range from 1.673 to 2.173 GeV. The relative phases between φ and Λ(1520) production amplitudes were obtained in the kinematic region where the two resonances overlap. The measurement results support strong constructive interference when K+K- pairs are observed at forward angles but destructive interference for proton emission at forward angles. Furthermore, the observed interference effect does not account for the s=2.1 GeV bump structure in forward differential cross sections for φ photoproduction. This fact suggests possible exotic structures such as a hidden-strangeness pentaquark state, a new Pomeron exchange, or rescattering processes via other hyperon states.