This paper explores the interactions of a groupwork team composed of both local and exchange students, with heterogeneous competence in English, in an English-medium CLIL context at a technical university in Catalonia. Plurilingual and multimodal conversation analysis is used to trace how the students jointly complete an academic task. The research conducted specifically analyses how students categorise themselves and each other in terms of their expertise, and the procedures and resources the students deploy to accomplish the task. The data show that participants' heterogeneous linguistic repertoires are not an obstacle for successfully completing the task, for constructing subject knowledge, or for establishing a climate of mutual understanding and cooperation. The analysis refers to the tension emerging in the data between the interactional principles of progressivity --actions oriented towards task completion--and intersubjectivity--actions oriented towards resolving communicative difficulties. It also focuses on how co-participants mobilise diverse resources from their communicative repertoires, including plurilingual resources, gesture and material artefacts, in managing the task. The main argument put forward is that in instructional environments in which students are expected to build subject matter knowledge using languages that they are simultaneously learning (e.g. CLIL), considering their whole communicative repertoires as valuable resources for their learning is a promising approach.