We consider finite deformations of the Hull-Strominger system. Starting from the heterotic superpotential, we identify complex coordinates on the off-shell parameter space. Expanding the superpotential around a supersymmetric vacuum leads to a third-order Maurer-Cartan equation that controls the moduli. The resulting complex effective action generalises that of both Kodaira-Spencer and holomorphic Chern-Simons theory. The supersymmetric locus of this action is described by an L 3 algebra. C The off-shell N = 1 parameter space and holomorphicity of Ω 41 C.1 SU(3) × SO(6) structures in the NS-NS sector 41 C.2 SU(3) × SO(6 + n) structures in heterotic supergravity 44 C.3 The off-shell hermitian structure on V 45 D Comments on D-terms 48 D.1 Massless deformations 48 D.2 Including bundle moduli 52 D.3 Polystable bundles 54 D.4 Full Maurer-Cartan equations 56 E Massless moduli 571 More precisely, it is the complex vector bundle V C (defined in appendix C.3) that is a holomorphic bundle. 2 The curvature R in the Bianchi identity is the curvature of a connection on T X, satisfying its own hermitian Yang-Mills conditions in order for the equations of motion to be fulfilled [33]. To O(α ′ ), this connection is ∇ − , given by taking the connection in (A.7) with the opposite sign for H.