= Solution
Use a real <scalar field> and Minkowski <action> $S[\phi]$, with the vacuum selected by a <Feynman propagator> pole prescription or adiabatic vacuum boundary conditions. The normalized <correlation functions> are
$$
G_n(x_1,\ldots,x_n)=\frac{\int\mathcal D\phi\,\phi(x_1)\cdots\phi(x_n)e^{iS[\phi]}}{\int\mathcal D\phi\,e^{iS[\phi]}}=\langle\Omega|\mathcal T\{\widehat\phi_H(x_1)\cdots\widehat\phi_H(x_n)\}|\Omega\rangle.
$$
Here $|\Omega\rangle$ is the interacting vacuum and $\mathcal T$ denotes <time ordering>. These are the canonical <vacuum expectation values>, rather than unordered products of noncommuting fields. The <functional integral> formula assumes a regulated measure and the same vacuum prescription on both sides.
Couple an <external source> $J$ linearly to the <scalar field> and choose the <normalized vacuum generating functional>
$$
Z[J]=\frac{\int\mathcal D\phi\,\exp\left(iS[\phi]+i\int d^dx\,J\phi\right)}{\int\mathcal D\phi\,e^{iS[\phi]}},\qquad Z[0]=1.
$$
Then <functional derivatives> insert the fields:
$$
G_n=\left.i^{-n}\frac{\delta^nZ}{\delta J(x_1)\cdots\delta J(x_n)}\right|_{J=0},\qquad Z[J]=\sum_{n=0}^\infty\frac{i^n}{n!}\int G_n(x_1,\ldots,x_n)\prod_{j=1}^n J(x_j)\,d^dx_j.
$$
In the convention used below, the <connected generating functional> is
$$
\boxed{Z[J]=e^{iW[J]},\qquad W[J]=-i\log Z[J],\qquad G_n^c=\left.i^{1-n}\frac{\delta^nW}{\delta J(x_1)\cdots\delta J(x_n)}\right|_{J=0}.}
$$
The logarithm removes products of disconnected components. For example the <connected correlation function> $G_2^c=G_2-G_1G_1$; at three points one subtracts all three $G_2G_1$ products and adds $2G_1G_1G_1$. Normalization has already removed <vacuum bubbles>.
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