In a search for $omega$ mesic states, the production of $omega$-mesons in coincidence with forward going protons has been studied in photon induced reactions on $^{12}$C for incident photon energies of 1250 - 3100 MeV. The $pi^0 gamma$ pairs from decays of bound or quasi-free $omega$-mesons have been measured with the CBELSA/TAPS detector system in coincidence with protons registered in the MiniTAPS forward array. Structures in the total energy distribution of the $pi^0 gamma$ pairs, which would indicate the population and decay of bound $omega~^{11}$B states, are not observed. The $pi^0 gamma$ cross section of 0.3 nb/MeV/sr observed in the bound state energy regime between -100 and 0 MeV may be accounted for by yield leaking into the bound state regime because of the large in-medium width of the $omega$-meson. A comparison of the measured total energy distribution with calculations suggests the real part $V_0$ of the $omega~^{11}$B potential to be small and only weakly attractive with $V_0(rho=rho_0) = -15pm$ 35(stat) $pm$20(syst) MeV in contrast to some theoretical predictions of attractive potentials with a depth of 100 - 150 MeV.
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