  We present a construction of regular Stein neighborhoods of a union of
maximally totally real subspaces $M=(A+iI)\mathbb{R}^n$ and $N=\mathbb{R}^n$ in
$\mathbb{C}^n$, provided that the entries of a real $n \times n$ matrix $A$ are
sufficiently small. Our proof is based on a local construction of a suitable
plurisubharmonic function $\rho$ near the origin, such that the sublevel sets
of $\rho$ are strongly pseudoconvex and admit strong deformation retraction to
$M\cup N$. We also give the application of this result to totally real
immersions of real $n$-manifolds in $\mathbb{C}^n$ with only finitely many
double points, and such that the union of the tangent spaces at each
intersection in some local coordinates coincides with $M\cup N$, described
above.
