An experimental search for quasi-real photoproduction of possible Theta(+) and Theta(++) resonances was performed using the decay modes pKs and pK(+) with the HERMES experiment at DESY. We report evidence for the existence of a Theta(+) peak with a statistical significance of 4 sigma. The absence of a peak in the pK(+) spectrum in the vicinity of the Theta(+) mass yields an estimate of the upper limit for the quasi-real photoproduction cross-section of the Theta(++) resonance. In addition, a search for exotic baryon resonances with S = -2, Q = -2 or 0 has been performed in quasi-real photoproduction oil a deuterium target by isolating the decay channel Xi(-) pi(-) -> Lambda pi(-)pi(-) -> p pi(-)pi(-)pi(-) and Xi(-)pi(+) -> Lambda pi(-)pi(+) -> p pi(-)pi(-)pi(+). No evidence for the previously reported Xi(--)(1860) and Xi(0)(1860) resonances is found in the Xi(-)pi(-) and Xi(-)pi(+) invariant mass spectra, respectively. Upper limits for their photoproduction cross sections of 2.1 nb and 2.5 nb are found at the 90% confidence level. The photoproduction cross section for the Xi(0)(1530) resonance, which has the same decay mode as the hypothetical Xi(0)(1860) peak, is found to be between 9 and 24 nb.