We study ac demagnetization in frustrated arrays of single-domain ferromagnetic islands, exhaustively resolving every (Ising-like) magnetic degree of freedom in the systems. Although the net moment of the arrays is brought near zero by a protocol with sufficiently small step size, the final magnetostatic energy of the demagnetized array continues to decrease for finer-stepped protocols and does not extrapolate to the ground-state energy. The resulting complex disordered magnetic state can be described by a maximum-entropy ensemble constrained to satisfy just nearest-neighbor correlations.