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Transformer-based models have demonstrated superior performance in various fields, including natural language processing and computer vision. However, their enormous model size and high demands in computation, memory, and communication limit their deployment to edge platforms for local, secure inference. Binary transformers offer a compact, low-complexity solution for edge deployment with reduced bandwidth needs and acceptable accuracy. However, existing binary transformers perform inefficiently on current hardware due to the lack of binary specific optimizations. To address this, we introduce COBRA, an algorithm-architecture co-optimized binary Transformer accelerator for edge computing. COBRA features a real 1-bit binary multiplication unit, enabling matrix operations with -1, 0, and +1 values, surpassing ternary methods. With further hardware-friendly optimizations in the attention block, COBRA achieves up to 3,894.7 GOPS throughput and 448.7 GOPS/Watt energy efficiency on edge FPGAs, delivering a 311× energy efficiency improvement over GPUs and a 3.5× throughput improvement over the state-of-the-art binary accelerator, with only negligible inference accuracy degradation.