  We measured missing mass spectrum of the $^{12}{\rm C}(\gamma,p)$ reaction
for the first time in coincidence with potential decay products from $\eta'$
bound nuclei. We tagged an ($\eta+p$) pair associated with the $\eta'N\to\eta
N$ process in a nucleus. After applying kinematical selections to reduce
backgrounds, no signal events were observed in the bound-state region. An upper
limit of the signal cross section in the opening angle $\cos\theta^{\eta
p}_{lab}<-0.9$ was obtained to be 2.2 nb/sr at the 90$\%$ confidence level. It
is compared with theoretical cross sections, whose normalization ambiguity is
suppressed by measuring a quasifree $\eta'$ production rate. Our results
indicate a small branching fraction of the $\eta'N\to\eta N$ process and/or a
shallow $\eta'$-nucleus potential.
