The question on the existence of nontrivial pseudocharacters on anomalous products of locally indicable groups is considered. Some generalizations of theorems of R. I. Grigorchuk and V. G. Bardakov on the existence of nontrivial pseudocharacters on free products with the amalgamation subgroup are found. It is proved that they exist on an anomalous product 〈 G, x | w = 1〉, where G is a locally indicable noncyclic group. We also prove some other propositions on the existence of nontrivial pseudocharacters on anomalous products of groups. Results on the second cohomologies of these products and their nonamenability follow from the propositions on the existence of nontrivial pseudocharacters on these groups.