Tensor algebra involving multiple sparse operands is severely memory bound, making it a challenging target for acceleration. Furthermore, irregular sparsity complicates traditional techniques-such as tiling-for ameliorating memory bottlenecks. Prior sparse tiling schemes are sparsity unaware: they carve tensors into uniform coordinate-space shapes, which leads to low-occupancy tiles and thus lower exploitable reuse. To address these challenges, this paper proposes dynamic reflexive tiling (DRT), a novel tiling method that improves data reuse over prior art for sparse tensor kernels, unlocking significant performance improvement opportunities. DRT's key idea is dynamic sparsity-aware tiling. DRT continuously re-tiles sparse tensors at runtime based on the current sparsity of the active regions of all input tensors, to maximize accelerator buffer utilization while retaining the ability to co-iterate through tiles of distinct tensors. Through an extensive evaluation over a set of SuiteSparse matrices, we show how DRT can be applied to multiple prior accelerators with different dataflows (ExTensor, OuterSPACE, MatRaptor), improving their performance (by 3.3×, 5.1× and 1.6×, respectively) while adding negligible area overhead. We apply DRT to higher-order tensor kernels to reduce DRAM traffic by 3.9× and 16.9× over a CPU implementation and prior-art tiling scheme, respectively. Finally, we show that the technique is portable to software, with an improvement of 7.29× and 2.94× in memory overhead compared to untiled sparse-sparse matrix multiplication (SpMSpM).