The magnetization and magnetically induced elastic strains of rare-earth Ising antiferromagnets DyAlO3 and TbAlO3 are shown to exhibit an unusual behavior associated with low-temperature metamagnetic phase transitions. As an external magnetic field is applied and then removed slowly, the state of the magnetic system in these compounds follows quite different paths on the H-T diagram. Small alternating-sign variations in the field magnitude cause the magnetic system to switch reversibly from one path to another, which is accompanied by sharp changes in the magnetization and elastic strains. The observed anomalies are shown to be due to the magnetization process being quasi-adiabatic in character in the compounds under study. This fact should be taken into account in interpreting the data on the magnetization and magnetostriction in Ising antiferromagnets undergoing metamagnetic transitions at low temperatures. Experimentally, quasi-adiabatic magnetization makes it possible to determine the critical fields for metamagnetic transitions very exactly and to investigate the H-T phase diagram at temperatures that are far below the minimum temperature of a helium bath and are unattainable under strictly isothermal conditions.