How do I customize columns when searching to add related records

When adding a product to a record, there is no easy way to customize which columns are shown, and the only way to search is to use the product name.

Luckily for me, I get the columns I need and I can find the product just fine. But some other users don't get the same columns and I haven't found a way to customize which columns are found in these search results.

In the parent record, I can customize the columns and that seems to apply globally, but for search results... nope

Is there any way to work around or to actually achieve this?

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