  In Dubouloz and Mangolte (Fake real planes: exotic affine algebraic models of
$\mathbb{R}^{2}$, arXiv:1507.01574, 2015) we define and partially classify fake
real planes, that is, minimal complex surfaces with conjugation whose real
locus is diffeomorphic to the euclidean real plane $\mathbb{R}^{2}$.
Classification results are given up to biregular isomorphisms and up to
birational diffeomorphisms. In this note, we describe in an elementary way
numerous examples of fake real planes and we exhibit examples of such planes of
every Kodaira dimension $\kappa\in \{-\infty,0,1,2\}$ which are birationally
diffeomorphic to $\mathbb{R}^{2}$.
