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Distributed fiber-optic sensing (DFOS) technologies such as Distributed Acoustic Sensing (DAS) generate continuous, high-resolution spatio-temporal data at rates exceeding hundreds of megabytes per second, creating severe challenges for real-time storage and transmission. While neural compression methods can achieve strong rate-distortion performance on sensing data, prior approaches either require per-segment retraining incompatible with streaming, or have only been validated offline. None have been demonstrated end-to-end on a continuously streaming DAS acquisition system at scale. We present STeP-C (Spatio-Temporal Predictive Compression), a neural compression framework that decouples spatial encoding, temporal prediction, and entropy-optimized residual coding, trained end-to-end with a rate-distortion objective motivated by conditional entropy reduction. Critically, we identify entropy coding as the dominant bottleneck for streaming deployment. We address it through GPU-side symbol computation, cached probability tables, and parallel GIL-free arithmetic coding, together with FP16 inference and batched entropy coding, enabling a full real-time pipeline built around a GPU ring buffer. We validate this pipeline on a live DAS testbed, achieving a 98.85% file size reduction at 31.04 dB full-band PSNR and an average throughput of 535 MB/s, exceeding the 324 MB/s requirement for 1 kHz, 81 km DAS streams. We further show that STeP-C substantially outperforms conventional codecs at practical streaming bitrates, can be applied to unseen channel ranges within the same acquisition system without retraining, and largely maintains downstream detector consistency, with 96.62% high-confidence class agreement and 100% Jaccard similarity on detected event types.