Missing data in Gibbs sampling for dynamic linear models

Suppose I have the following DLM:

$x_t = \Phi x_{t-1} + w_t$

$y_t = A x_t + v_t$

$x_0 \sim N(\mu_0,\Sigma_0)$

$w_t \sim N(0,Q)$

$v_t \sim N(0,R)$

Let $\Theta = \{\mu_0,\Sigma_0,\Phi,Q,A,R\}$. I can do Gibbs sampling for the joint distribution $P(x,\Theta|y)$ by sampling from the conditionals $P(x|\Theta,y)$ and $P(\Theta|x,y)$.

As I understand it, for $P(x|\Theta,y)$, sampling is done via the Forward Filtering Backward Sampling algorithm, which translates to running a Kalman smoother, which handles partially and fully missing values in $y_t$ by more or less ignoring them in the calculations.

On the other hand, for sampling $P(\Theta|x,y)$, how should I go about handling missing values in $y_t$? Can I ignore fully missing $y_t$ when calculating the full conditionals? What about partially missing $y_t$? Or should I simulate missing $y_t$ in another Gibbs step?