Identifying unauthenticated referred users

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Hi All, 

I'm new to shopify apps/apis and have spent a couple days digging through docs and this forum but can't quite find exactly what I'm after. I wanted to check if this is a possibility and maybe if someone could point me In the right direction or where I should start/look;

I want to be able to send a user that is known/authenticated on a non-shopify site to a shopify store with some unique identifying string. I then want to use this unique string and a shopify app to identify them with the external site, such that I can modify the shopify store pages during their visit (because they're authenticated with external site). All without authenticating them on the shopify/store side?

Does that make sense? Is this a pipe dream?

Thank you in advance for your time.