Can I construct a complete (as a Boolean algebra) $\aleph_0$ saturated elementary extension of a given Boolean algbera?

Question

This is a follow-up to my previous question:

Let $B$ be an arbitrary Boolean algebra. Can one construct a $\aleph_0$-saturated $B^* \succ B$ that is complete, i.e., all joins and meets exist in $B^*$?

For my previous question, which involved uncountable cardinal numbers instead of $\aleph_0$, the answer is no. What happens we only need countable saturation instead?

Edit: This is also a [reference-request] post.