Given a family of Boolean algebras $\mathcal{B}=\{B_i\colon i\in I\}$ with respective Stone spaces $S_i$, the algebra of clopen (both closed and open) subsets of the product space $\textstyle\prod_{i\in I}S_i$ is called the *free product* of $\mathcal B$. This algebra is typically denoted by $\textstyle\bigotimes_{i\in I}B_i$ (and I will use the standard "tensor" notation for finite free products in the obvious manner).

I am interested in the (possible) Boolean algebras which admit only very particular decompositions in terms of the free product.

Is there an uncountable Boolean algebra $B$ such that if $B$ is isomorphic to $A\otimes C$ then either $A$ or $C$ is countable?