Using computer simulations, we study a membrane of parallel narrow pores filled with one-dimensional wires of hydrogen-bonded water molecules. We show that such a membrane is equivalent to a system of effective charges located at opposite sides of the membrane offering a computationally efficient way to model correlation effects in water-filled nanopore membranes. Based on our simulations we predict that membranes with square pore lattices undergo a continuous order-disorder transition to an antiferroelectric low-temperature phase in which water wires in adjacent pores are oriented in opposite directions. Strong antiferroelectric correlations exist also in the disordered phase far above the critical temperature or in membranes with geometric frustration, leading to a dielectric constant that is reduced considerably with respect to the case of uncoupled water wires. These correlations are also expected to hinder proton translocation through the membrane.