Te/Sb/Ge/Ag (TAGS) materials with rather high concentrations of cation vacancies exhibit improved thermoelectric properties as compared to corresponding conventional TAGS (with constant Ag/Sb ratio of 1) due to a significant reduction of the lattice thermal conductivity. There are different vacancy ordering possibilities depending on the vacancy concentration and the history of heat treatment of the samples. In contrast to the average alpha-GeTe-type structure of TAGS materials with cation vacancy concentrations directions but not equidistantly spaced, comparable to the nanostructures of compounds (GeTe)(n)Sb2Te3. Upon heating, the nanostructures transform into long-periodically ordered trigonal phases with parallel van der Waals gaps. These phases are slightly affected by stacking disorder but distinctly different from the alpha-GeTe-type structure reported for conventional TAGS materials. Deviations from this structure type are evident only from HRTEM images along certain directions or very weak intensities in diffraction patterns. At temperatures above similar to 400 degrees C, a rock-salt-type high-temperature phase with statistically disordered cation vacancies is formed. Upon cooling, the long-periodically trigonal phases are reformed at the same temperature. Quenched nanostructured Ge0.53Ag0.13Sb0.27 square Te-0.07(1) and Ge0.61Ag0.11Sb0.22 square Te-0.06(1) exhibit ZT values as high as 1.3 and 0.8, respectively, at 160 degrees C, which is far below the phase transition temperatures. After heat treatment, i.e., without pronounced nanostructure and when only reversible phase transitions occur, the ZT values of Ge0.53Ag0.13Sb0.27 square Te-0.07(1) and Ge0.61Ag0.11Sb0.22 square Te-0.06(1) with extended van der Waals gaps amount to 1.6 at 360 degrees C and 1.4 at 410 degrees C, respectively, which is at the top end of the range of high-performance TAGS materials.