Reinforcement Learning from Verifiable Rewards (RLVR) has significantly improved the reasoning capabilities of large language models (LLMs), particularly in multi-turn agentic settings involving environment interaction like tool use. However, fine-tuning such models remains prohibitively expensive due to high computational requirements, limiting accessibility. We propose MARLaaS (Multi-tenant Asynchronous RL as a Service), a system for concurrent RL fine-tuning across multiple users and tasks. Our approach is based on two key ideas: (1) sharing a base model across tenants using lightweight LoRA adapters, and (2) a disaggregated asynchronous architecture that decouples rollout generation, environment interaction, and policy training into independently scheduled stages. This design enables tasks to progress through the RL pipeline at their own pace in an event-driven manner, reducing cross-task interference, idle time, and end-to-end latency. In multi-task settings (we report up to 32 concurrent tasks), MARLaaS achieves single-task state-of-the-art performance while improving accelerator utilization by up to 4.3x and reducing end-to-end training time by 85%.