How can I run an exact match search?
I encountered this "limitation" with matrix search pages and multiple-text-fields.
Yes, setting word logic on ALL words won't match on individual items in your multiple-text-field, as it would have to match the entire contents of your multiple-text-field to work.
Setting Any word logic will return "partial" results as you have found.
A Squiz employee may suggest later that you could go all out and use funnelback for this - but that is overkill.
My filtering solution involved using Any word logic and a REST JS asset nested in your search page default format. Nested div is passed %list_current_asset_assetid% as a GET variable.
You can use a keyword modifier "as_asset" on this %list_current_asset_assetid% GET variable to get the contents of the related assets metadata field for that asset, and then do a comparison against the asset_id of the current page and then only print or use the ones that match. You may need to split the multiple-text-field metadata into an array to use with a for loop.
Instead of using a search page, could you use an asset listing, and pass the contents of the multiple text field as a dynamic parameter for list of assetid's? You can use a list of assetid's instead of a 'root' asset. Have a dynamic variable set that gets it from the current asset, metadata keyword, that should be able to be nested directly into the paint layout instead of the search page.
Working from memory here, I think multiple text might by default return list separated by ';', but I think you can use ^replace to change to commas, then just pass it in to the asset listing. Wouldn't work for anything else, but since it's assetid's anyway, should be able to use an asset listing.
EDIT: Upon re-reading I think I misunderstood the structure. My way would work if asset A is displaying the list of related assets, and asset A's metadata has the list of assets that it is related to. Now that I look at it a second time, looks like you are defining the relationship the other direction, hence the need for a search. Forcing quotes around the search term perhaps, to delineate a whole word, rather than a partial?