Embedding-as-a-Service (EaaS) is an effective and convenient deployment solution for addressing various NLP tasks. Nevertheless, recent research has shown that EaaS is vulnerable to model extraction attacks, which could lead to significant economic losses for model providers. For copyright protection, existing methods inject watermark embeddings into text embeddings and use them to detect copyright infringement. However, current watermarking methods often resist only a subset of attacks and fail to provide \textit{comprehensive} protection. To this end, we present the region-triggered semantic watermarking framework called RegionMarker, which defines trigger regions within a low-dimensional space and injects watermarks into text embeddings associated with these regions. By utilizing a secret dimensionality reduction matrix to project onto this subspace and randomly selecting trigger regions, RegionMarker makes it difficult for watermark removal attacks to evade detection. Furthermore, by embedding watermarks across the entire trigger region and using the text embedding as the watermark, RegionMarker is resilient to both paraphrasing and dimension-perturbation attacks. Extensive experiments on various datasets show that RegionMarker is effective in resisting different attack methods, thereby protecting the copyright of EaaS.