We construct 1-parameter families of non-periodic embedded minimal surfaces of infinite genus in T × R, where T denotes a flat 2-tori. Each of our families converges to a foliation of T × R by T. These surfaces then lift to minimal surfaces in R 3 that are periodic in horizontal directions but not periodic in the vertical direction. In the language of crystallography, our construction can be interpreted as disordered stacking of layers of periodically arranged catenoid necks. Our work is motivated by experimental observations of twinning defects in periodic minimal surfaces, which we reproduce as special cases of stacking disorder.