The practical deployment of diffusion-based Neural Video Compression (NVC) faces critical challenges, including severe information loss, prohibitive inference latency, and poor temporal consistency. To bridge this gap, we propose DiffVC-RT, the first framework designed to achieve real-time diffusion-based perceptual NVC. First, we introduce an Efficient and Informative Model Architecture. Through strategic module replacements and pruning, this architecture significantly reduces computational complexity while mitigating structural information loss. Second, to address generative flickering artifacts, we propose Explicit and Implicit Consistency Modeling. We enhance temporal consistency by explicitly incorporating a zero-cost Online Temporal Shift Module within the U-Net, complemented by hybrid implicit consistency constraints. Finally, we present an Asynchronous and Parallel Decoding Pipeline incorporating Mixed Half Precision, which enables asynchronous latent decoding and parallel frame reconstruction via a Batch-dimension Temporal Shift design. Experiments show that DiffVC-RT achieves 80.1% bitrate savings in terms of LPIPS over VTM-17.0 on HEVC dataset with real-time encoding and decoding speeds of 206 / 30 fps for 720p videos on an NVIDIA H800 GPU, marking a significant milestone in diffusion-based video compression.