We present STREAMRHF, an unsupervised anomaly detection algorithm for data streams. Our algorithm builds on some of the ideas of Random Histogram Forest (RHF), a state-of-the-art algorithm for batch unsupervised anomaly detection. STREAMRHF constructs a forest of decision trees, where feature splits are determined according to the kurtosis score of every feature. It irrevocably assigns an anomaly score to data points, as soon as they arrive, by means of an incremental computation of its random trees and the kurtosis scores of the features. This allows efficient online scoring and concept drift detection altogether. Our approach is tree-based which boasts several appealing properties, such as explainability of the results. We conduct an extensive experimental evaluation on multiple datasets from different real-world applications. Our evaluation shows that our streaming algorithm achieves comparable average precision to RHF while outperforming state-of-the-art streaming approaches for unsupervised anomaly detection with furthermore limited computational complexity.