Robustness of Pisot-regular sequences

We consider numeration systems based on a d-tuple U=(U1,…,Ud) of sequences of integers and we define (U,K)-regular sequences through K-recognizable formal series, where K is any semiring. We show that, for any d-tuple U of Pisot numeration systems and any semiring K, this definition does not depend on the greediness of the U-representations of integers. The proof is constructive and is based on the fact that the normalization is realizable by a 2d-tape finite automaton. In particular, we use an ad hoc operation mixing a 2d-tape automaton and a K-automaton in order to obtain a new K-automaton.