Generative models enhance neuroimaging through data augmentation, quality improvement, and rare condition studies. Despite advances in realistic synthetic MRIs, evaluations focus on texture and perception, lacking sensitivity to crucial anatomical fidelity. This study proposes a new metric, called WASABI (Wasserstein-Based Anatomical Brain Index), to assess the anatomical realism of synthetic brain MRIs. WASABI leverages \textit{SynthSeg}, a deep learning-based brain parcellation tool, to derive volumetric measures of brain regions in each MRI and uses the multivariate Wasserstein distance to compare distributions between real and synthetic anatomies. Based on controlled experiments on two real datasets and synthetic MRIs from five generative models, WASABI demonstrates higher sensitivity in quantifying anatomical discrepancies compared to traditional image-level metrics, even when synthetic images achieve near-perfect visual quality. Our findings advocate for shifting the evaluation paradigm beyond visual inspection and conventional metrics, emphasizing anatomical fidelity as a crucial benchmark for clinically meaningful brain MRI synthesis. Our code is available at https://github.com/BahramJafrasteh/wasabi-mri.