= Solution
Assume a real <refractive index> and write the <scattering potential> as $V=\langle V\rangle+W$, so $\langle W\rangle=0$ by construction. With the real kernel components in the question, the logarithmic <Rytov approximation> separates into
$$
\operatorname{Re}\chi_1=-\int a(\mathbf r,\mathbf r')V(\mathbf r')\,d\mathbf r',\qquad \operatorname{Im}\chi_1=-\int b(\mathbf r,\mathbf r')V(\mathbf r')\,d\mathbf r'.
$$
Include the incident <wave phase> and the mean-potential <wave phase> in the deterministic reference $\phi_0$: at each observation point it is $\arg\psi_i-\int b\langle V\rangle$. It need not be spatially constant. The <amplitude> is $|\psi_i|\exp[-\int aV]$. The <phase covariance in the first Rytov approximation> therefore starts from the fluctuating <wave phase>
$$
\boxed{\varphi(\mathbf r)=-\int b(\mathbf r,\mathbf r')W(\mathbf r')\,d\mathbf r'.}
$$
Under the integrability assumptions needed to interchange the expectation and integral,
$$
\boxed{\langle\varphi(\mathbf r)\rangle=-\int b(\mathbf r,\mathbf r')\langle W(\mathbf r')\rangle\,d\mathbf r'=0.}
$$
For a complex absorbing potential, the corresponding phase fluctuation is $-\int[b\operatorname{Re}W+a\operatorname{Im}W]$; the displayed scalar formula is the real-index case. The zero mean comes from centering $V$, not from setting the mean of $V$ equal to zero. In particular the printed $\langle n\rangle=0$ does not imply $\langle V\rangle=0$. A physical positive <refractive index> usually has a nonzero background mean; a zero-mean assumption normally refers to its fluctuation. The algebra above remains meaningful for a signed real random field and explicitly retains its mean <scattering potential>.
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