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

Fractions p/q ∈ [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 10ⁿ-1. A number of results generalise Midy's theorem to expansions of 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 R\'enyi. First we define the Midy property for a general real base β >1 and derive a necessary condition for validity of the Midy property. For β =12(1+√5) we characterize prime denominators q, which satisfy the property.