Suppose that $V$ is a model of $\sf ZFC$, and fix some regular $\kappa$, say $\omega_1$ for practical purposes.

Let $\cal U$ be an ultrafilter on $\omega_1$ in $V$ which is non-principal and even uniform. Let $\Bbb P$ be a forcing such that:

- $\Bbb P$ does not add bounded subsets to $\kappa$, but its generic defines a canonical unbounded subset of $\kappa$, say $G$.
- $\Bbb P$ satisfies $\kappa^+$-c.c.
- $\Bbb P$ is homogeneous.

Of course that $\cal U$ is not an ultrafilter in $V[G]$ since it is no longer closed under supersets. But we can ask whether or not $\mathcal U\cup\{G\}$ is an ultrafilter base.

Is there a condition on $\cal U$ (or an additional condition on $\Bbb P$ or $G$) which guarantee that $\mathcal U\cup\{G\}$ is an ultrafilter base in $V[G]$?