Let’s say I’m doing A/B testing, and this one particular user gets randomly assigned into group B for example. I have a choice right away to either store that on their account in my database, or on their browser. Then tomorrow, that same person visits the website from another device, and randomly gets group A assigned, and sees the A-version of the website. And then, from this new device where they were viewing version A so far, they now sign into their account.
Would it be correct to now load everything B, which has potentially been stored on their account? Or would it be correct to persist the current visit, and keep showing A in this local browser? Or, go so far with persisting the new visit to actually store this new version A onto their account for any further page renders from their first device.
This is all assuming that the different versions of the site differ visually, and significantly, so, what the user expects is therefore quite important.