  This paper introduces a generalization of the ddc-condition for complex
manifolds. Like the dd^c-condition, it admits a diverse collection of
characterizations, and is hereditary under various geometric constructions.
Most notably, it is an open property with respect to small deformations. The
condition is satisfied by a wide range of complex manifolds including all
compact complex surfaces, and all compact Vaisman manifolds. We show there are
computable invariants of a real homotopy type which in many cases prohibit it
from containing any complex manifold satisfying such ddc-type conditions in low
degrees. This gives rise to numerous examples of almost complex manifolds which
cannot be homotopy equivalent to any of these complex manifolds.
