Cell-free massive multiple-input multiple-output (MIMO) is a promising technology for next-generation communication systems. This work proposes a novel partially coherent (PC) transmission framework to cope with the challenge of phase misalignment among the access points (APs), which is important for unlocking the full potential of cell-free massive MIMO technology. With the PC operation, the APs are only required to be phase-aligned within clusters. Each cluster transmits the same data stream towards each user equipment (UE), while different clusters send different data streams. We first propose a novel algorithm to group APs into clusters such that the distance between two APs is always smaller than a reference distance ensuring the phase alignment of these APs. Then, we propose new algorithms that optimize the combining at UEs and precoding at APs to maximize the downlink sum data rates. We also propose a novel algorithm for data stream allocation to further improve the sum data rate of the PC operation. Numerical results show that the PC operation using the proposed framework with a sufficiently small reference distance can offer a sum rate close to the sum rate of the ideal fully coherent (FC) operation that requires network-wide phase alignment. This demonstrates the potential of PC operation in practical deployments of cell-free massive MIMO networks.