We investigated the $\bar{K}$-nucleus interaction through the simultaneous measurement of the inclusive $^{12}{\rm C}(K^-, p)$ and exclusive $K^{-}$-escape $^{12}{\rm C}(K^-, p K^-_{esc})$ reactions at $1.8$ GeV/$c$ at J-PARC. The present measurement explicitly focuses on the $K^{-}$ escape process for the first time, successfully accomplishing a direct experimental determination of the imaginary part of the $K^{-}$ optical potential. The differential cross section for the $K^{-}$-escape reaction was determined to be $436\pm 6(\mathrm{\text{stat.}})\pm 44(\mathrm{\text{syst.}})\mu \mathrm{\text{b/sr}}$. A simultaneous likelihood fit yielded real and imaginary potential strengths of $V_0=-72{}_{-5}^{+3}(\mathrm{\text{stat.}}){}_{-8}^{+0}(\mathrm{\text{syst.}})\mathrm{\text{MeV}}$ and $W_0=-100{}_{-1}^{+7}(\mathrm{\text{stat.}}){}_{-16}^{+0}(\mathrm{\text{syst.}})\mathrm{\text{MeV}}$ at the nuclear center, respectively. The derived $W_0$ is significantly stronger than that predicted by theoretical models based on one-nucleon processes, suggesting possible contribution of multi-nucleon involving processes.