  We develop a theory of complex Kuranishi structures on projective schemes.
These are sufficiently rigid to be equivalent to weak perfect obstruction
theories, but sufficiently flexible to admit global complex Kuranishi charts.
  We apply the theory to projective moduli spaces M of stable sheaves on
Calabi-Yau 4-folds. Using real derived differential geometry, Borisov-Joyce
produced a virtual homology cycle on M. In the prequel to this paper we
constructed an algebraic virtual cycle on M. We prove the cycles coincide in
homology after inverting 2 in the coefficients. And when Borisov-Joyce's real
virtual dimension is odd, their virtual cycle is 2-torsion.
