Two-dimensional arrays of magnetically coupled nanomagnets provide a mesoscopic platform for exploring collective phenomena as well as realizing a broad range of spintronic devices. In particular, the magnetic coupling plays a critical role in determining the nature of the cooperative behavior and providing new functionalities in nanomagnet-based devices. Here, we create coupled Ising-like nanomagnets in which the coupling between adjacent nanomagnetic regions can be reversibly converted between parallel and antiparallel through solid-state ionic gating. This is achieved with the voltage-control of the magnetic anisotropy in a nanosized region where the symmetric exchange interaction favors parallel alignment and the antisymmetric exchange interaction, namely the Dzyaloshinskii-Moriya interaction, favors antiparallel alignment of the nanomagnet magnetizations. Applying this concept to a two-dimensional lattice, we demonstrate a voltage-controlled phase transition in artificial spin ices. Furthermore, we achieve an addressable control of the individual couplings and realize an electrically programmable Ising network, which opens up new avenues to design nanomagnet-based logic devices and neuromorphic computers.