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In the appropriate limit, a type IIB string theory setup involving D3 branes,wrapped D5 branes, and fluxes on a conifold generally leads to a supergravitybackground involving a warped version of the conifold with fluxes. We study thesupergravity dual of the baryonic branch of the Klebanov Strassler theory bywriting down a very general conifold metric--the non-K\"ahler resolvedwarped-deformed conifold--and a general set of fluxes that satisfy thesupergravity equations of motion, and derive the necessary constraints thatallow the geometry to be dual to an $\mathcal{N}=1$ supersymmetric gauge theoryin $3+1$ dimensions. These backgrounds encompass known solutions, such as theKS, MN and Butti et al. models, but the added layer of generality can lead to alarger class of gauge-gravity dualities. We also present many consistencychecks that validate our background matches known cases for certain values inour parameter space. This is a companion paper to arXiv:1805.03676 [hep-th]covering the section 'IR physics, dualities, and supersymmetry'.

A companion to "Non-K\"ahler Deformed Conifold, Ultra-Violet Completion and Supersymmetric Constraints in the Baryonic Branch"