This letter proposes a multi-stream selection framework for Cell-Free Multiple-Input Multiple-Output (CF-MIMO) networks. Partially coherent transmission has been considered by clustering Access Point ss (APs) into phase-aligned clusters to address the challenges of phase misalignment and inter-cluster interference. A novel stream selection algorithm is developed to dynamically allocate multiple streams to each multi-antenna User Equipment (UE), ensuring that the system optimizes the sum rate while minimizing inter-cluster and inter-stream interference. Numerical results validate the effectiveness of the proposed method in enhancing spectral efficiency and fairness in distributed CF-MIMO networks.