Lagrangian skeleta of very affine complete intersections

Let $Z^{\circ }$ be a complete intersection inside $({\mathbb{C}}^{\ast }{)}^n$ that compactifies to a smooth Calabi-Yau subvariety $Z$ inside a Fano toric variety $X$. We compute the skeleton of $Z^{\circ }$ and describe its decomposition into standard pieces that are mirror to toric varieties, which generalises the existing results in the case of hypersurfaces. This set-up was first considered by Batyrev and Borisov, who used combinatorial techniques to construct a mirror pair $(Z,\check{Z})$ of such complete intersections. We use our main result to establish homological mirror symmetry for Batyrev-Borisov pairs in the large-volume limit.