  We consider the Einstein-Maxwell-fluid constraint equations, and make use of
the conformal method to construct and parametrize constant-mean-curvature
hyperboloidal initial data sets that satisfy the shear-free condition. This
condition is known to be necessary in order that a spacetime development admit
a regular conformal boundary at future null infinity. We work with initial data
sets in a variety of regularity classes, primarily considering those data sets
whose geometries are weakly asymptotically hyperbolic, as defined in
[arXiv:1506.03399]. These metrics are $C^{1,1}$ conformally compact, but not
necessarily $C^2$ conformally compact. In order to ensure that the data sets we
construct are indeed shear-free, we make use of the conformally covariant
traceless Hessian introduced in [arXiv:1506.03399]. We furthermore construct a
class of initial data sets with weakly asymptotically hyerbolic metrics that
may be only $C^{0,1}$ conformally compact; these data sets are insufficiently
regular to make sense of the shear-free condition.
