The race to heuristically solve non-deterministic polynomial-time (NP) problems through efficient methods is ongoing. Recently, optics was demonstrated as a promising tool to find the ground-state of a spin-glass Ising Hamiltonian, which represents an archetypal NP problem. However, achieving completely programmable spin couplings in these large-scale optical Ising simulators remains an open challenge. Here, by exploiting the knowledge of the transmission matrix of a random medium, we experimentally demonstrate the possibility of controlling the couplings of a fully-connected Ising spin-system. By tailoring the input wavefront we showcase the possibility of modifying the Ising Hamiltonian both by accounting for an external magnetic-field and by controlling the number of degenerate ground-states and their properties and probabilities. Our results represent a relevant step toward the realisation of fully-programmable Ising-machines on thin optical-platforms, capable of solving complex spin-glass Hamiltonians on a large scale.