  In a recent series of papers, new exact analytical solutions to field
equations of General Relativity representing interior spacetimes sourced by
stationary rigidly rotating cylinders of fluids with various equations of state
have been displayed. This work is currently extended to the case of
differentially rotating irrotational fluids. The results are presented in a new
series of papers considering, in turn, a perfect fluid source, arXiv:2305.11565
[gr-qc], as well as the three anisotropic pressure cases already studied in the
rigidly rotating configuration. The axially directed pressure case has already
been developed in arXiv:2307.07263. Here, a fluid with an azimuthally directed
pressure is considered. A general method for generating the corresponding new
mathematical solutions to the field equations when the ratio
$h=$pressure/energy density varies with the radial coordinate is proposed, and
a class of solutions exemplifying this recipe is derived. Then, the case where
$h=const.$ is solved. It splits into two subclasses depending on the value of
$h$. The mathematical and physical properties of these three classes are
analyzed which provides some constraints on $h$, different for each class and
subclass. Their matching to an exterior Lewis-Weyl vacuum and the conditions
for avoiding an angular deficit are discussed. A comparison with the rigidly
rotating fluid case is provided.
