Generating long and stylized human motions in real time is critical for applications that demand continuous and responsive character control. Despite its importance, existing streaming approaches often operate directly in the raw motion space, leading to substantial computational overhead and making it difficult to maintain temporal stability. In contrast, latent-space VAE-Diffusion-based frameworks alleviate these issues and achieve high-quality stylization, but they are generally confined to offline processing. To bridge this gap, LILAC (Long-sequence Incremental Low-latency Arbitrary Motion Stylization via Streaming VAE-Diffusion with Causal Decoding) builds upon a recent high-performing offline framework for arbitrary motion stylization and extends it to an online setting through a latent-space streaming architecture with a sliding-window causal design and the injection of decoded motion features to ensure smooth motion transitions. This architecture enables long-sequence real-time arbitrary stylization without relying on future frames or modifying the diffusion model architecture, achieving a favorable balance between stylization quality and responsiveness as demonstrated by experiments on benchmark datasets. Supplementary video and examples are available at the project page: https://pren1.github.io/lilac/