In general, the product of two Hermitian operators $\phi$ will not be Hermitian, unless the two operators commute.
Question: is $X = T \phi(t_1) \phi(t_2)$ Hermitian? It doesn't seem to be if
$T \phi_1 \phi_2 = \phi_1\phi_2\theta(t_1-t_2) + \phi_2\phi_1\theta(t_2-t_1).$ (Weinberg)
If not, how can we interpret $\langle 0 |X|0\rangle$ as the vacuum correlation function, if X is not an observable?