Cooperation is fundamental to human societies, and one of the important paths for its emergence and maintenance is reciprocity. In prisoner's dilemma (PD) experiments, reciprocal strategies are often effective at attaining and maintaining high cooperation. In many public goods (PG) games or n-person PD experiments, however, reciprocal strategies are not successful at engendering cooperation. In the present paper, we attribute this difficulty to a coordination problem against free riding among reciprocators: Because it is difficult for the reciprocators to coordinate their behaviors against free riders, this may lead to inequality among players, which will demotivate them from cooperating in future rounds. We propose a new mechanism, institutionalized reciprocity (IR), which refers to embedding the reciprocal strategy as an institution (i.e., institutionalizing the reciprocal strategy). We experimentally demonstrate that IR can prevent groups of reciprocators from falling into coordination failure and achieve high cooperation in PG games. In conclusion, we argue that a natural extension of the present study will be to investigate the possibility of IR to serve as a collective punishment system.