The British actor surfaced Friday near a coffee shop in Studio City, where he appeared to be on his own and minding his business -- but by the same token, he wasn't really going incognito ... while JA was rocking a ball cap and shades, he removed both.

What this seems to mean ... Joe wasn't shy about having his mug out there, and he didn't appear to mind being photographed either. The dude's just out there doing his thing.

This is interesting for a few different reasons -- one, he's technically on Taylor's turf here, and not just that ... but she's out here in L.A. right now with none other than TK -- who's been a man about town as well. Of course, if these 2 were to run into each other ... awk.

Taylor and Joe were together for 6 years before they split up in 2023 -- and in the aftermath of her getting with Travis ... she's got a new album coming out that seems primed to take JA to task, at least that's what the Swifties seem to think anyway.

In any case ... Joe doesn't seem to care if he crosses paths with T-Swift and Travis -- 'cause, frankly, he actually has a reason to be in town, and it's work-related.

If you didn't hear, Emma Stone is gonna be in this new movie with the same director who just helped win her an Oscar ... and Joe's gonna be in it too. In fact, this will mark the second time that Joe has worked with ES on a Yorgos Lanthimos movie ... they already did "The Favourite" together a few years back, and YL is reteaming with these two again for his latest.

Interestingly enough ... Taylor and Emma have a good friendship as it is, and now -- Emma's slated to work with her ex, who's now cruising around L.A. as Tay Tay is on a lil' break.