Suppose I am trying to make inference about a parameter $\mu$. I have a prior $$ \mu \sim N(0,\sigma^2), $$ and I observe two correlated signals about $\mu,$ namely $x_1, x_2$ where $$ \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = N\left(\begin{pmatrix} \mu \\ \mu \end{pmatrix}, \begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \sigma_2^2 \end{pmatrix}\right), $$ so $x_1$ and $x_2$ are just "noised up" $\mu$. One can show that the posterior for $\mu$ conditional on $x_1$ and $x_2$ is given by $$ \mu \mid x_1, x_2 \sim N\left(\frac{\tau_1x_1 - \rho\sqrt{\tau_1\tau_2}(x_1+x_2) + \tau_2x_2}{\tau_1 - 2\rho\sqrt{\tau_1\tau_2} + \tau_2 + (1-\rho^2)\tau_\mu}, \frac{1-\rho^2}{\tau_1 - 2\rho\sqrt{\tau_1\tau_2} + \tau_2 + (1-\rho^2)\tau_\mu}\right), $$ where $\tau$ is the precision. I want to know how "informative" $x_1$ and $x_2$ are as a function of $\rho$. I'm going to define the measure of informativeness as the expected reduction in the mean-squared error of our estimate for $\mu$.* Because we want to minimize the MSE, we'll take the estimator for $\mu$ to be the posterior mean (see here).
Now, my intuition (which I guess is wrong) would be that independent signals would be the most informative about the true value of $\mu$ since correlated signals contain "overlapping" information. But we see that as $\lvert\rho\rvert\to1$ the variance on the right hand side goes to $0,$ in other words, the more correlated the signals are, the better our posterior seems to be. What's going on here? Is my intuition wrong, or is my interpretation of the result wrong?
The best "secondary" intuition I can come up with is that perhaps the correlation is allowing us to pinpoint $\mu$. I know that as $\lvert \rho \rvert \to 1,$ the values of $x_2$ become concentrated around the line $x_2 = \mu + \tfrac{\sigma_2}{\sigma_1}(x_1-\mu)$ (see here), so of course if we observed both $x_1$ and $x_2$ we would be able to back out the value of $\mu$ from this.
Finally, if this is true (i.e. that correlated signals here lead to more accurate estimates), what is the minimal change we could make to the setup so that more correlation leads to a lower quality posterior estimate for $\mu$?
*If this notion of informativeness seems unintuitive to you, the same conclusions can be drawn if we consider instead the mutual information between $\mu$ and $x_1, x_2$.