= Solution
With the stated flat priors, the full joint density, up to a constant, is
$$
\boxed{
\mathbf1_{\{\sigma^2>0,\tau^2>0\}}
\prod_{i=1}^N
N(x_i\mid\xi_i,\sigma_{x,i}^2)
N(y_i\mid\eta_i,\sigma_{y,i}^2)
N(\eta_i\mid\alpha+\beta\xi_i,\sigma^2)
N(\xi_i\mid\mu,\tau^2).}
$$
The priors on $\alpha,\beta,$ and $\mu$ contribute constants on $\mathbb R$, while those on the two variances contribute constants on $(0,\infty)$. These are <improper priors>, so posterior propriety must be checked; the full-rank, sufficiently large-data case used below is proper.
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