Generative AI has advanced capabilities, enabling these systems to participate as teammates in human teams. Yet, the potential consequences of including an AI teammate for team climate have yet to be explored. Thus, we investigate how shared decisional ownership between humans and AI, as well as the perception of AI as a teammate affect team climate (including its subdimensions). We conducted an experiment with 85 participants in 35 teams collaborating with a generative AI teammate on a team decision-making task. We demonstrate that human decisional ownership improves team climate, while AI decisional ownership has a non-significant negative impact. However, when AI is perceived as a teammate, its decisional ownership also enhances team climate. The qualitative analysis provides additional insights into how these perceptions emerge. Our findings provide a nuanced understanding of the mechanisms of team-AI collaboration that shape team climate and offer practical guidance for fostering a positive team climate.