Midy's Theorem in non-integer bases and divisibility of Fibonacci numbers

Fractions $\frac{p}{q}\in [0,1)$ with prime denominator $q$ written in decimal have a curious property described by Midy’s Theorem, namely that two halves of their period (if it is of even length $2n$) sum up to ${10}^n-1$. A number of results generalise Midy’s theorem to expansions of $\frac{p}{q}$ in different integer bases, considering non-prime denominators, or dividing the period into more than two parts. We show that a similar phenomena can be studied even in the context of numeration systems with non-integer bases, as introduced by Rényi. First we define the Midy property for a general real base $\beta >1$ and derive a necessary condition for validity of the Midy property. For $\beta =\frac{1}{2}(1+\sqrt{5})$ we characterize prime denominators $q$, which satisfy the property.