Magnetosheath jets represent localized enhancements in dynamic pressure observed within the magnetosheath. These energetic entities, carrying excess energy and momentum, can impact the magnetopause and disrupt the magnetosphere. Therefore, they play a vital role in coupling the solar wind and terrestrial magnetosphere. However, our understanding of the morphology and formation of these complex, transient events remains incomplete over two decades after their initial observation. Previous studies have relied on oversimplified assumptions, considering jets as elongated cylinders with dimensions ranging from $0.1\, R_{\rm E}$ to $5\, R_{\rm E}$ (Earth radii). In this study, we present simulation results obtained from Amitis, a high-performance hybrid-kinetic plasma framework (particle ions and fluid electrons) running in parallel on graphics processing units (GPUs) for fast and more environmentally friendly computation compared to CPU-based models. Considering realistic scales, we present the first global, three-dimensional (3D in both configuration and velocity spaces) hybrid-kinetic simulation results of the interaction between solar wind plasma and the Earth. Our high-resolution kinetic simulations reveal the 3D structure of magnetosheath jets, showing that jets are far from being simple cylinders. Instead, they exhibit intricate and highly interconnected structures with dynamic 3D characteristics. As they move through the magnetosheath, they wrinkle, fold, merge, and split in complex ways before a subset reaches the magnetopause.