Corruption in public procurement transforms state institutions into private entities where public resources get diverted for the benefit of a few. On this matter, much of the discussion centers on the legal fulfillment of the procurement process, while there are fewer formal analyses related to the corporate features which are most likely to signal organized crime and corruption. The lack of systematic evidence on this subject has the potential to bias our understanding of corruption, making it overly focused on the public sector. Nevertheless, corruption scandals worldwide tell of the importance of taking a better look at the misuse and abuse of corporations for corrupt purposes. In this context, the research presented here seeks to contribute to the understanding of the criminal conspiracy of companies involved in public procurement corruption scandals under a network and complexity science perspective. To that end, we make use of a unique dataset of the corporate ownership and management information of four important and recently documented cases of corruption in Mexico, where hundreds of companies were used to embezzle billions of dollars. Under a bipartite network approach, we explore the relations between companies and their personnel (shareholders, legal representatives, administrators, and commissioners) in order to characterize their static and dynamic networked structure. In terms of organized crime and using different network properties, we describe how these companies connect with each other due to the existence of shared personnel with role multiplicity, leading to very different conspiracy networks. To best quantify this behavior, we introduce a heuristic network-based conspiracy indicator that together with other network metrics describes the differences and similarities among the networks associated with each corruption case. Finally, we discuss some public policy elements that might be needed to be considered in anti-corruption efforts related to corporate organized crime.