Stein neighborhood bases for complex--totally real unions

We study Stein neighborhood bases for unions of closed complex and totally real submanifolds, with particular attention to a linear configuration with noncompact intersection. We prove that the set $X=\{w=0\}\cup \{\mathrm{\Im }z=\mathrm{\Im }w=0\}\subset {\text{C}}^2$ admits a Stein neighborhood basis. The proof starts from the explicit plurisubharmonic defining function $\rho (z,w)=(\mathrm{\Im }w{)}^2+{\sinh }^2(\mathrm{\Im }z)\mathrm{\mid }w{\mathrm{\mid }}^2$ and obtains the necessary non-uniform control at infinity by adding a locally uniformly convergent series of nonnegative plurisubharmonic terms constructed by Oka–Weil approximation. We also give an example of a closed complex submanifold and a closed totally real submanifold in ${\text{C}}^2$ with compact intersection whose union has no Stein neighborhood basis, showing that the unrestricted union problem has a negative answer.