  In this paper we study the expansions of real numbers in positive and
negative real base as introduced by R\'enyi, and Ito & Sadahiro, respectively.
In particular, we compare the sets $\mathbb{Z}_\beta^+$ and
$\mathbb{Z}_{-\beta}$ of nonnegative $\beta$-integers and $(-\beta)$-integers.
We describe all bases $(\pm\beta)$ for which $\mathbb{Z}_\beta^+$ and
$\mathbb{Z}_{-\beta}$ can be coded by infinite words which are fixed points of
conjugated morphisms, and consequently have the same language. Moreover, we
prove that this happens precisely for $\beta$ with another interesting
property, namely that any integer linear combination of non-negative powers of
the base $-\beta$ with coefficients in $\{0,1,\dots,\lfloor\beta\rfloor\}$ is a
$(-\beta)$-integer, although the corresponding sequence of digits is forbidden
as a $(-\beta)$-integer.
