  We study the quantum spin liquid phase in a variant of the Kitaev model where
the bonds of the honeycomb lattice are distributed in a Kekul\'e pattern. The
system supports gapped and gapless Z_2 quantum spin liquids with interesting
differences from the original Kitaev model, the most notable being a gapped Z_2
spin liquid on a Kagome lattice. Perturbing the exactly solvable model with
antiferromagnetic Heisenberg perturbations, we find a magnetically ordered
phase stabilized by a quantum `order by disorder' mechanism, as well as an
exotic continuous phase transition between the topological spin liquid and this
magnetically ordered phase. Using a combination of field theory and Monte-Carlo
simulations, we find that the transition likely belongs to the 3D-XYxZ_2
universality class.
