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Classical lagrangians of fermions are always constructed out of Grassmann numbers. No exception. In both of OP's cases, the mass term is nonvanishing: In the first case the mass term is proportional to $\psi_1^*\psi_2^*-\psi_2^*\psi_1^* = 2\psi_1^*\psi_2^* \neq 0.$ In the second case, I write in the chiral basis: $$\gamma^0=\begin{pmatrix}& \sigma^0 ...

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