Temporal graphs are critical for modeling dynamic systems where interactions evolve over time, with a central challenge being the characterization of structural cohesion. The temporal edge-core, defined under a temporal proximity constraint Δ, quantifies the stability and density of connections within subgraphs and is essential for applications such as anomaly detection and information diffusion. Existing edge-core decomposition methods, however, are designed for static graphs and are computationally prohibitive in streaming environments due to frequent edge arrivals and deletions. We present TECM, an efficient framework for streaming temporal edge-core decomposition that leverages the localized impact of edge updates within Δ-incident neighbors. TECM incrementally updates core values through Δ-aware traversals and localized H-index analysis, and incorporates batch processing to handle high-velocity streams. Extensive experiments on real and synthetic temporal networks demonstrate that TECM delivers speedups of several orders of magnitude over state-of-the-art static baselines, providing a scalable and principled solution for real-time structural analysis in evolving temporal graphs.