Recently I had an issue regarding a site that runs 2 stores, but shared the same ROOT category between the stores, but had seperate categories per store (categories are enabled at store level)
The problem is that the integrated search service ( Algolia ) woudl index all products to all sites. The issue is that you could then search products in one store from another. This is not wnated, as they do have seperate products.
The only reason they sahre a root catalog is that all products do actually belong to both stores, which facilitates the ability to swap between stores, and retain products in cart from each store. So in effect you can checkout in either store with another stores products.
Search, however shoudl not be cross-site
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