Leveraging the spatiotemporal continuous aware ness of low Earth orbit satellites, space-based spectrum monitoring systems can construct a spectrum situation map (SSM) to facilitate spectrum surveillance and management in mobile cognitive communication networks. However, due to limited on orbit processing capabilities and unfavorable ground-to-satellite transmission environments, spectrum measurements will inevitably be incomplete and corrupted by anomalies. Existing SSM construction methods assume stationary spectral environments and do not consider the dynamic changes in spectrum situation. Considering spectrum state is evolving constantly over time, this paper proposes a robust time-aware online streaming tensor (R TAST) completion algorithm by exploiting the time-frequency space correlation and temporal properties in real-world spectral measurements. Based on compressed wideband sampling, the proposed R-TAST algorithm integrates rank estimation, anomaly removal, and dynamic spectral tensor completion to achieve spectrum situation completion and evolutionary prediction. Numerical analyses conducted on simulated and realistic ray tracing based datasets demonstrate the effectiveness and efficiency of the proposed R-TAST in comparison with state-of-the-art streaming tensor completion and prediction algorithms.