= Solution
Define the prior-predictive nuisance integrals
$$
f_A(\theta)=\int L_A(\theta,\boldsymbol\alpha)
\pi(\boldsymbol\alpha)\,d\boldsymbol\alpha,
\qquad
f_B(\theta)=\int L_B(\theta,\boldsymbol\beta)
\pi(\boldsymbol\beta)\,d\boldsymbol\beta.
$$
Then
$$
\mathcal P_{AB}(\theta)\mathcal Z_{AB}
=f_A(\theta)f_B(\theta)\pi(\theta).
$$
For each individual analysis, <Bayes theorem> also gives
$$
f_A(\theta)=\frac{\mathcal Z_A\mathcal P_A(\theta)}{\pi(\theta)},
\qquad
f_B(\theta)=\frac{\mathcal Z_B\mathcal P_B(\theta)}{\pi(\theta)}.
$$
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