  Flat surface phases are unstable during growth and known to become rough.
This does not exclude the possibility that surface reconstruction order
persists in rough growing surfaces, in analogy with so-called equilibrium
reconstructed rough phases. We investigate this in the context of KPZ type
dynamics, using the restricted solid on solid model with negative mono-atomic
step energies. Long range reconstruction order is strictly speaking absent in
the thermodynamic limit, but the reconstruction domain walls become trapped at
surface ridge lines, and the reconstruction order parameter fluctuates
critically with the KPZ dynamic exponent at finite but large length scales.
