Quantum key distribution (QKD) provides information-theoretic security rooted in quantum physics, while high-dimensional (HD) encoding increases both noise tolerance and secret-key yield. Multicore fibers (MCFs), a leading platform for next-generation telecom networks, are a natural substrate for HD-QKD. Field demonstrations over deployed MCFs have so far relied on a hybrid qudit encoding strategy that combines two path (core modes) with the time-bin photonic degree of freedom, rather than exploiting the full set of available core modes. Although practical, this current approach incurs intrinsic efficiency penalties that grow with dimension. Encoding a qudit in the full set of core modes removes these penalties, but it requires the coherent control of all cores at once and has so far been restricted to laboratory settings. Here we implement a four-dimensional ($d=4$) QKD protocol that directly exploits the full set of core modes of a four-core MCF, operating over an installed MCF network across the Universidad de Concepci'on campus under continuous environmental perturbations. We further benchmark the scheme using superconducting nanowire detectors at $10$dB channel loss, achieving a composable finite-key rate of $R=6.19\times {10}^{-3}$bits/pulse, the highest per-pulse rate reported to date for deployed HD-QKD at comparable loss. This result establishes core-mode encoding as a viable architecture for realistic, high-rate quantum-secure communications.