On regular Stein neighborhoods of a union of two maximally totally real subspaces in $$\mathbb {C}^n$$

We construct regular Stein neighborhoods of a union of two maximally totally real subspaces \(M=(A+iI)\mathbb {R}^n\) and \(N=\mathbb {R}^n\) in \(\mathbb {C}^n\), provided that the eigenvalues of the real \(n \times n\) matrix A are sufficiently small. This result is applied to provide regular Stein neighborhoods of an immersed totally real n-manifold in a complex n-manifold, with only finitely many double points, and such that the union of the tangent spaces at each double point in some local coordinates coincides with \(M\cup N\), described above.