The first measurement of diffractive scattering of quasi-real photons with large momentum transfer γp→γY, where Y is the proton dissociative system, is made using the H1 detector at HERA. The measurement is performed for initial photon virtualities Q2<0.01 GeV2. Single differential cross sections are measured as a function of W, the incident photon–proton centre of mass energy, and t, the square of the four-momentum transferred at the proton vertex, in the range 175<W<247 GeV and 4<|t|<36 GeV2. The W dependence is well described by a model based on perturbative QCD using a leading logarithmic approximation of the BFKL evolution. The measured |t| dependence is harder than that predicted by the model and those observed in exclusive vector meson production.