This article focuses on the tensions that occur when project actors work towards digitally driven transformation. Based on the notion of “digital ambidexterity” - defined as the organisational capability to successfully enable the exploitation of existing digital technologies and exploration of new digital initiatives - the study aims to understand how project actors navigate ambidexterity tensions. The empirical part consists of a qualitative case study of an inter-organisational construction project involving about 15 industry actors, with Skanska as the main contractor. Based on public case data and about 15 in-depth interviews with key representatives from involved actors, the article show how digital ambidexterity is navigated through two parallel practices; one is driven by the navigating mindset of each project actor reflecting the operating logic of their home organisations, and one driven by actors in the organisational setting having to deal with sudden unexpected deviations. It is concluded that to navigate digital ambidexterity better, firms should anchor technological benefits in effectiveness measures (doing the right things for the project) and not only efficiency improvements (doing more with less).