Complex conjugate of fermionic terms in zero-dimensional Ginzburg-Landau theory formulation

Reading a section of the book Mirror Symmetry (section 9.6) on zero-dimensional Landau-Ginzberg Theory and we are asked to consider the action

$S(z,\bar{z},\psi_1,\bar{\psi_1},\psi_2,\bar{\psi_2})=|\partial W|^2 -\partial^2W \psi_1\psi_2 -\overline{(\partial^2W)}\bar{\psi_1}\bar{\psi_2}$

While trying to rationalize why we chose this specific function, it's not difficult to see the parallels between what the book had discussed before (the same thing but without the complex pairs), but I was a little confused about the sign of the last term. In my head, it would make sense to add the complex conjugate of the term $-\partial^2 W\psi_1\psi_2$ in order to make the action real. But up till now, I learned that

$-\overline{(\partial^2W\psi_1\psi_2)}=-\overline{(\partial^2W)}\bar{\psi_2}\bar{\psi_1}=\overline{(\partial^2W)}\bar{\psi_1}\bar{\psi_2}$

So the last term differs by a negative factor from what I would expect. I was hoping to get clarification on where this discrepancy is coming from (maybe we're not adding the complex conjugate at all?). Thanks.