Recent studies on counterfeiting in a monetary search framework show that counterfeiting does not occur in a monetary equilibrium. These findings are inconsistent with the observation that counterfeiting of bank notes has been a serious problem in some countries. In this paper, we show that counterfeiting can exist as an equilibrium outcome in a model in which money is not perfectly recognizable and thus can be counterfeited. A competitive search environment is employed in which sellers post offers and buyers direct their search based on posted offers. When sellers are uninformed about the quality of the money, their offers are pooling and thus buyers can extract rents by using counterfeit money. In this case, counterfeit notes can coexist with genuine notes under certain conditions. We also explicitly model the interaction between sellers' verification decisions and counterfeiters' choices of counterfeit quality. This allows us to better understand how policies can affect counterfeiting.