The investigation of interior spacetimes sourced by stationary cylindrical anisotropic fluids is pursued and specialized here to rigidly rotating fluids with an azimuthally directed pressure. Based on the occurrence of an extra degree of freedom in the equations, two general methods for constructing different classes of exact solutions to the field equations are proposed. Exemplifying such recipes, a bunch of solutions are constructed. Axisymmetry and regularity conditions on the axis are examined, and the spacetimes are properly matched to a vacuum exterior. A number of classes and subclasses are thus studied, and an analysis of their features leads to sorting out three classes whose appropriate mathematical and physical properties are discussed. This work is part of a larger study of the influence of anisotropic pressure in general relativity, using cylindrical symmetry as a simplifying assumption, and considering, in turn, each principal stress direction. It has been initiated in companion Papers I and II, where the pressure was assumed to be axially directed, and is followed by Paper IV considering radial pressure and Paper V contrasting the previous results with the corresponding dust and perfect fluid solutions.

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