We construct the string duals of the defect theories generated when NfNf flavor D5-branes intersect NcNc color D3-branes along a 2+12+1 dimensional subspace. We work in the Veneziano limit in which NcNc and NfNf are large and Nf/NcNf/Nc is fixed. By smearing the D5-branes, we find supergravity solutions that take into account the backreaction of the flavor branes and preserve two supercharges. When the flavors are massless the resulting metric displays an anisotropic Lifshitz-like scale invariance. The case of massive quarks is also considered.