The structures of RNA sequences play a vital role in various cellular processes, while existing genomic foundation models (FMs) have struggled with precise sequence-structure alignment, due to the complexity of exponential combinations of nucleotide bases. In this study, we introduce OmniGenome, a foundation model that addresses this critical challenge of sequence-structure alignment in RNA FMs. OmniGenome bridges the sequences with secondary structures using structure-contextualized modeling, enabling hard in-silico genomic tasks that existing FMs cannot handle, e.g., RNA design tasks. The results on two comprehensive genomic benchmarks show that OmniGenome achieves state-of-the-art performance on complex RNA subtasks. For example, OmniGenome solved 74% of complex puzzles, compared to SpliceBERT which solved only 3% of the puzzles. Besides, OmniGenome solves most of the puzzles within $1$ hour, while the existing methods usually allocate $24$ hours for each puzzle. Overall, OmniGenome establishes wide genomic application cases and offers profound insights into biological mechanisms from the perspective of sequence-structure alignment.