  I construct a generalisation of Mantovan's almost product structure to
Shimura varieties of Hodge type with hyperspecial level structure at $p$ and
deduce that the perfection of the Newton strata are pro-\'etale locally
isomorphic to the perfection of the product of a central leaf and a
Rapoport-Zink space. The almost product formula can be extended to obtain an
analogue of Caraiani's and Scholze's generalisation of the almost product
structure for Shimura varieties of Hodge type.
