We complete our recent classification [GMS11] of compact inner symmetric
spaces with weakly complex tangent bundle by filling up a case which was left open, and
extend this classification to the larger category of compact homogeneous spaces with positive
Euler characteristic. We show that a simply connected compact equal rank homogeneous
space has weakly complex tangent bundle if and only if it is a product of compact equal
rank homogeneous spaces which either carry an invariant almost complex structure (and are
classified by Hermann [H56]), or have stably trivial tangent bundle (and are classified by
Singhof and Wemmer [SW86]), or belong to an explicit list of weakly complex spaces which
have neither stably trivial tangent bundle, nor carry invariant almost complex structures
