Advances in omics technology have enriched the understanding of the biological mechanisms of diseases, which has provided a new approach for cancer research. Multi-omics data contain different levels of cancer information, and comprehensive analysis of them has attracted wide attention. However, limited by the dimensionality of matrix models, traditional methods cannot fully use the key high-dimensional global structure of multi-omics data. Moreover, besides global information, local features within each omics are also critical. It is necessary to consider the potential local information together with the high-dimensional global information, ensuring that the shared and complementary features of the omics data are comprehensively observed. In view of the above, this article proposes a new tensor integrative framework called the strong complementarity tensor decomposition model (BioSTD) for cancer multi-omics data. It is used to identify cancer subtype specific genes and cluster subtype samples. Different from the matrix framework, BioSTD utilizes multi-view tensors to coordinate each omics to maximize high-dimensional spatial relationships, which jointly considers the different characteristics of different omics data. Meanwhile, we propose the concept of strong complementarity constraint applicable to omics data and introduce it into BioSTD. Strong complementarity is used to explore the potential local information, which can enhance the separability of different subtypes, allowing consistency and complementarity in the omics data to be fully represented. Experimental results on real cancer datasets show that our model outperforms other advanced models, which confirms its validity.