In this paper we study the expansions of real numbers in positive andnegative 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 ofconjugated morphisms, and consequently have the same language. Moreover, weprove that this happens precisely for $\beta$ with another interestingproperty, namely that any integer linear combination of non-negative powers ofthe base $-\beta$ with coefficients in $\{0,1,\dots,\lfloor\beta\rfloor\}$ is a$(-\beta)$-integer, although the corresponding sequence of digits is forbiddenas a $(-\beta)$-integer.
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