We introduce the definition of tensorized block rational Krylov subspaces and their relation with multivariate rational functions, extending the formulation of tensorized Krylov subspaces introduced in [2]. Moreover, we develop methods for the solution of tensor Sylvester equations with low multilinear or Tensor Train rank, based on projection onto a tensor block rational Krylov subspace. We provide a convergence analysis and some strategies for poles selection based on the techniques developed in [1]. References [1] A. A. Casulli and L. Robol. “An effcient block rational Krylov solver for Sylvester equations with adaptive pole selection”. In: arXiv preprint arXiv:2301.08103 (2023). [2] D. Kressner and C. Tobler. “Krylov subspace methods for linear systems with tensor product structure”. In: SIAM Journal on Matrix Analysis and Applications 31.4 (2010), pp. 1688–1714