  We present a procedure for asymptotic gluing of hyperboloidal initial data
sets that preserves the shear-free condition. Our construction is modeled on a
previous gluing construction by the last three named authors, but with
significant modifications that incorporate the shear-free condition. We rely on
the special H\"older spaces, and the corresponding theory for elliptic
operators on weakly asymptotically hyperbolic manifolds, introduced by the
authors and applied to the Einstein constraint equations in two previous
papers.
