We consider N=(2,0) backgrounds of IIB supergravity on eight-manifolds M_8 with strict SU(4) structure. We give the explicit solution to the Killing spinor equations as a set of algebraic relations between irreducible su(4) modules of the fluxes and the torsion classes of M_8. One consequence of supersymmetry is that M_8 must be complex. We show that the conjecture of arxiv:1010.5789 concerning the correspondence between background supersymmetry equations in terms of generalized pure spinors and generalized calibrations for admissible static, magnetic D-branes, does not capture the full set of supersymmetry equations. We identify the missing constraints and express them in the form of a single pure-spinor equation which is well defined for generic SU(4)\times SU(4) backgrounds. This additional equation is given in terms of a certain analytic continuation of the generalized calibration form for codimension-2 static, magnetic D-branes.