With the rapid evolution of Large Language Models (LLMs), generative recommendation is gradually reshaping the paradigm of recommender systems. However, most existing methods remain confined to the interaction-driven next-item prediction paradigm, struggling to keep pace with the latest evolving trends or address the diverse recommendation tasks along with business-specific requirements in real-world scenarios. To this end, we present SIGMA, a Semantic-Grounded Instruction-Driven Generative Multi-Task Recommender deployed at AliExpress. Specifically, we first ground item entities in a unified latent space capturing both general semantics and collaborative signals. Building upon this, we introduce a hybrid item tokenization method for both precise modeling and efficient generation. Moreover, we construct a large-scale multi-task supervised fine-tuning dataset empowering SIGMA to fulfill various recommendation demands via instruction-following. Finally, we design a three-step item generation procedure integrated with an adaptive probabilistic fusion mechanism to calibrate the output distributions based on task-specific requirements for recommendation accuracy and diversity. Extensive offline experiments and online A/B tests demonstrate the effectiveness of SIGMA across various real-world recommendation tasks.