While trying to compute functional determinants, I faced an inconsistency, which I can exemplify by defining the following matrix
$$ M = \begin{pmatrix} i \frac{d}{dt} + t & 0 \\\ 0 & i \frac{d}{dt} - t\end{pmatrix} $$
Since $i \frac{d}{dt}$ is a hermitian operator, $M$ is a hermitian (infinite) matrix. Its determinant should be real. Then I can compute its determinant, using the fact that this matrix is block diagonal, being careful that they do not commute:
$$ \det M = \det \left( \left(i \frac{d}{dt} + t \right) \left(i \frac{d}{dt} - t \right) \right) = \det \left(-\left(\frac{d}{dt}\right)^2 - t^2 - i\right) $$
The commutation gave a factor $i$, and the operator $-\partial_t^2 - t^2 - i$ is no longer hermitian, and I believe the determinant is now a complex number.
Where am I wrong ?