Transnational cooperation in international corporations has become an important force in promoting the economic development of countries, and corruption in cross-cultural business has an important impact on the sustainable development of international cooperation. Based on the construction field, this study applies evolutionary game theory to the microlevel to investigate the corrupt behavior of international corporations from reputation perspectives, taking into account their reputation and cooperation behaviors. The findings indicate that the sensitivity of each party involved in the corruption behavior differs concerning international reputation, and a heightened reputation of the supervisory company can effectively curb the corrupt behavior of subcontracting. Additionally, the behavior of the general contracting company shows a sense of inertia, while the three main parties—general contracting company, supervisory company, and subcontracting company—exhibit multistage decision-making characteristics as their international reputation gradually improves. Through the lens of multinational enterprise cooperation and the development of the construction industry, this study aims to address the constraints faced by the construction industry in various countries and identify potential solutions. Furthermore, it provides insights into key issues related to international engineering corruption governance.