= Solution
Couple a real source through $H_J=H_0-\int d^Dx\,J(x)\phi(x)$. Its Fourier components obey $\widetilde J(-p)=\widetilde J(p)^*$, and the source term is $-\int_p\widetilde J(p)\widetilde\phi(p)^*$. Complete the square using the positive quadratic kernel from the previous part:
$$
H_J=\frac12\int_p\bigl(\widetilde\phi-\widetilde\Delta^{-1}\widetilde J\bigr)
\widetilde\Delta\bigl(\widetilde\phi-\widetilde\Delta^{-1}\widetilde J\bigr)^*
-\frac12\int_p\widetilde J\widetilde\Delta^{-1}\widetilde J^*.
$$
Translation of the integration variables in the regulated <Gaussian functional integral> therefore gives
$$
\boxed{Z_0[J]=Z_G\exp\left[\frac12\int_p\frac{|\widetilde J(p)|^2}{\alpha^{-1}p^2+m^2}\right].}
$$
This also fixes the sign: a real linear source increases the Gaussian generating function by a positive quadratic exponent, unlike an imaginary Fourier-characteristic-function source.
To specify $Z_G$ rather than conceal an infinite normalization, first impose finite volume and an <ultraviolet cutoff>, and choose orthonormal real coordinates for the independent field modes. If their number is $N_R$ and the eigenvalues of the quadratic form are $\Delta_j>0$, the measure $\prod_{j=1}^{N_R}d\phi_j$ gives
$$
\boxed{Z_G=\prod_{j=1}^{N_R}\left(\frac{2\pi}{\Delta_j}\right)^{1/2}
=(2\pi)^{N_R/2}(\det\Delta)^{-1/2}.}
$$
A measure $\prod_jd\phi_j/\sqrt{2\pi}$ removes the prefactor. In the continuum one may express this as $\log Z_G=-\tfrac12\operatorname{Tr}\log\Delta$ plus a source-independent measure constant, with the same regulator understood. The real-mode prescription prevents double counting the conjugate Fourier pairs. All normalized <correlation functions> follow from $Z_0[J]/Z_G$ and do not depend on that constant.
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