While generative models have set new benchmarks for Target Speaker Extraction (TSE), their inherent reliance on global context precludes deployment in real-time applications. Direct adaptation to streaming scenarios often leads to catastrophic inference performance degradation due to the severe mismatch between training and streaming inference. To bridge this gap, we present the first autoregressive (AR) models tailored for streaming TSE. Our approach introduces a Chunk-wise Interleaved Splicing Paradigm that ensures highly efficient and stable streaming inference. To ensure the coherence between the extracted speech segments, we design a historical context refinement mechanism that mitigates boundary discontinuities by leveraging historical information. Experiments on Libri2Mix show that while AR generative baseline exhibits performance degradation at low latencies, our approach maintains 100% stability and superior intelligibility. Furthermore, our streaming results are comparable to or even surpass offline baselines. Additionally, our model achieves a Real-Time-Factor (RTF) of 0.248 on consumer-level GPUs. This work provides empirical evidence that AR generative backbones are viable for latency-sensitive applications through the Chunk-wise Interleaved Splicing Paradigm.