Abstract. We consider the Einstein-Maxwell-fluid constraint equations, and make use of the conformal method to construct and parametrize constantmean-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; see [4]. 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 [1]. 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 [1]. 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.

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