In this paper, we study Multi- \({\mathcal {K}}\) -equivalence of multi-germs of functions on the plane, definable in a polynomially bounded o-minimal structure. As in Birbrair et al. (Annali SNS Pisa 17:81–92, 2017. https://doi.org/10.2422/2036-2145.201503_014), we partition the germ of the plane at origin into zones of arcs in such a way that it produces a non-Archimedean space (set of orders and width functions) compatible with a given multi-germ, encoding its asymptotic behaviour. Such a partition is called Multi-pizza. We show the existence, uniqueness and complete invariance of multi-pizzas with respect to the Multi- \({\mathcal {K}}\) -bi-Lipschitz equivalence.
