Continual instruction tuning (CIT) has emerged as a promising strategy for adapting large language models (LLMs) to new tasks while preserving historical knowledge. Most existing CIT methods have focused on offline CIT (offCIT), which assumes clearly defined task boundaries and allows multiple passes over the data. However, such assumptions rarely hold in real-world scenarios, where data arrive in a streaming fashion and task boundaries are unknown. This setting introduces critical challenges: the absence of task identifiers (task IDs), a significant imbalance in task-specific information, and inaccessibility to previously seen data. In this work, we propose Online Editing with Decoupled Implicit Task (OnEDIT), an online CIT(onCIT) approach to tackle these challenges. OnEDIT leverages a fixed-size adapter for the implicit task, balancing current and past knowledge through editing operations every time step without relying on task IDs or backpropagation. Extensive experiments on CIT benchmarks demonstrate that OnEDIT consistently maintains robust and stable performance, whereas existing state-of-the-art baselines often suffer from performance degradation in online settings. It suggests that OnEDIT achieves superior generalization across diverse task orders and model scales, while maintaining high efficiency and low memory overhead.