I have hundreds of products that share the same variants within their product pages and want to be able to search our inventory at our other locations of what is available.
EX: I will have a customer asking to see what else we have in Mens size 43 shoes at our other stores. If I search for the Mens 43 tag within Shopify (with inventory greater than zero), you can’t guarantee that what reveals will have available 43s because it’s possible that the product that shows up will have an available size, but not in the 43. It shows up because the 43 is contained in that product, NOT because it has an available 43.
Not all our products are online, so we can’t just send them to the website.
How do I exclusively search variants to help show my clients what we have available within the company not online?
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As you’ve found tags apply at the product level.
Ideally the information architecture on the merchant/staff side of things you’d be searching based on a sku convention to search variants either by exact sku or wildcard match. Or other relevant field like the variants full title which includes option values ie. Red / Small / Mens.
At least for the /admin product search you may be able to use search syntax to facilitate this
https://community.shopify.com/c/shopify-apis-and-sdks/graphql-api-search-products-variant-with-part-of-sku/td-p/664130
https://shopify.dev/api/usage/search-syntax
Afaik you cannot simply search by variant options through the variants themselves using the search syntax, nor can we use metafields for this(like if every variant has a unique non-sku code 
https://shopify.dev/api/admin-graphql/2022-07/objects/Product#field-product-options
https://shopify.dev/api/admin-graphql/2022-07/objects/ProductOption
https://shopify.dev/api/admin-graphql/2022-07/objects/SelectedOption
https://shopify.dev/api/admin-graphql/2022-07/objects/productvariant#field-productvariant-selectedoptions
There’s also using third party lowcode services to build more specific dashboards by working with the api.
Or shopify apps like usemehanic to build simple scripted tools, or frontend customizations.
You could sync the variants to a google sheet that had the relevant fields - size, inventory, etc. and then search in there? I could build out a workflow for you to do this using our app mesa if you’re interested.