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A $Q$-exact off-shell action is constructed for twisted abelian (2,0) theoryon a Lorentzian six-manifold of the form $M_{1,5} = C\times M_4$, where $C$ isa flat two-manifold and $M_4$ is a general Euclidean four-manifold. Theproperties of this formulation, which is obtained by introducing two auxiliaryfields, can be summarised by a commutative diagram where the Lagrangian and itsstress-tensor arise from the $Q$-variation of two fermionic quantities $V$ and$\lambda^{\mu\nu}$. This completes and extends the analysis in[arXiv:1311.3300].

Off-shell structure of twisted (2,0) theory