Our results concern the existence of a countable extension $\mathscr{U}$ of the countable atomless Boolean algebra $\mathscr{B}$ such that $\mathscr{U}$ is a "nonconstructive" extension of $\mathscr{B}$. It is known that for any fixed admissible indexing $\varphi$ of $\mathscr{B}$ there is a countable nonconstructive extension $\mathscr{U}$ of $\mathscr{B}$ (relative to $\varphi$). The main theorem here shows that there exists an extension $\mathscr{U}$ of $\mathscr{B}$ such that for any admissible indexing $\varphi$ of $\mathscr{B}$, $\mathscr{U}$ is nonconstructive (relative to $\varphi$). Thus, in this sense $\mathscr{U}$ is a countable totally nonconstructive extension of $\mathscr{B}$.