All countries recognize the right to adequate housing for all human beings. Yet, in many countries, social housing is in a critical state: most buildings need to be restored and better services should be guaranteed. Such actions should be part of a larger plan aimed to overcome the organizational and technological backwardness of the agencies that manage the social housing system. With a not large, but old public asset, the Italian context may represent an interesting case to start investigating difficulties and problems in the management of the social housing system that, as it occurs in most Mediterranean countries, arise when the public housing rental model is adopted. In the paper, a multiple case study on five Italian regional public Agencies responsible for the social housing system is discussed. In particular, the theoretical lens of process theory and ambidextrous business process management are adopted to study the Agencies’ “problematic situation” and identify innovative solutions to address it. The paper contributes to research and practice on process innovation and digital transformation of public administrations: three important lessons are derived and discussed also taking into consideration Industry 5.0, the vision on the future of industry recently proposed by the European Commission. Finally, the adoption of process theory combined with ambidextrous Business Process Management is an underexplored research method in the field of Architecture, Engineering, Construction, and Facility Management (AEC/FM) research. The results reported in the study reaffirm the potential deriving from its adoption also in such a field.