Uni-Stream brings streaming speech recognition and speech synthesis into one speech language model built on compact pretrained speech components. It processes interleaved speech and text streams and uses a two-stage training strategy to retain performance comparable to task-specific systems. The work also introduces a streaming-data annotation pipeline that produces transcriptions and word-level timestamps. Representation analysis shows that speech and text remain separated across modality-specific latent spaces, while later layers within each modality become increasingly specialized for recognition or synthesis.