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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 refers to the cached chords from the first phase rather than its previous predictions, which avoids the exposure bias problem. Since the input length is often constrained under real-time conditions, another potential problem is the loss of long-term sequential information. To make up for this disadvantage, we extract four musical features from a long-term music piece before the current time step as global information. In the experiment, we train SongDriver on some open-source datasets and an original àiSong Dataset built from Chinese-style modern pop music scores. The results show that SongDriver outperforms existing SOTA (state-of-the-art) models on both objective and subjective metrics, meanwhile significantly reducing the physical latency.