  We obtain a family of heterotic supergravity backgrounds describing
non-Kahler warped conifolds with three-form flux and an Abelian gauge bundle,
preserving N=1 supersymmetry in four dimensions. At large distance from the
singularity the usual Ricci-flat conifold is recovered. By performing a Z_2
orbifold of the T^{1,1} base, the conifold singularity can be blown-up to a
four-cycle, leading to a completely smooth geometry. Remarkably, the throat
regions of the solutions, which can be isolated from the asymptotic Ricci-flat
geometry using a double-scaling limit, possess a worldsheet CFT description in
terms of heterotic cosets whose target space is the warped resolved
orbifoldized conifold. Thus this construction provides exact solutions of the
modified Bianchi identity. By solving algebraically these CFTs we compute the
exact tree-level heterotic string spectrum and describe worldsheet
non-perturbative effects. The holographic dual of these solutions, in
particular their confining behavior, and the embedding of these fluxed
singularities into heterotic compactifications with torsion are also discussed.
