Real-time music accompaniment generation has a wide range of applications in the music industry, such as music education and live performances. However, automatic real-time music accompaniment generation is still understudied and often faces a trade-off between logical latency and exposure bias. In this paper, we propose SongDriver, a real-time music accompaniment generation system without logical latency nor exposure bias. Specifically, SongDriver divides one accompaniment generation task into two phases: 1) The arrangement phase, where a Transformer model first arranges chords for input melodies in real-time, and caches the chords for the next phase instead of playing them out. 2) The prediction phase, where a CRF model generates playable multi-track accompaniments for the coming melodies based on previously cached chords. With this two-phase strategy, SongDriver directly generates the accompaniment for the upcoming melody, achieving zero logical latency. Furthermore, when predicting chords for a timestep, SongDriver