TMZ's obtained footage of the couple strolling through the quaint alleyways of Capri Tuesday ... where they seem to have landed after a pit stop in Saint-Tropez over the weekend. As you can see, they were hand-in-hand ... with their team fully surrounding them.

They didn't let that kill the mood though ... Bennifer went about their business, taking in the sights and leisurely enjoying the city. Good view here too from the back ... just saying. 👀

Anyhoo ... eyewitnesses tell us it was quite the eventful day for them -- apparently, they stopped for a bite at a place called Ristorante Faraglioni ... but we're told there didn't seem to be a whole lot of eating. Our sources on the ground say it was mostly them hamming it up for paps.

We're also told they popped into a jewelry store at one point ... but it's unclear what, if anything, they bought. After that ... our sources say they hit up a vintage Chanel store. So, they're definitely in vacation mode -- shopping, chillin' and soaking in the atmosphere.

It seems their decision to go full PDA for the cams hasn't ceased either -- from land to sea, they're showing off their newfound love for everyone to witness. It'll be interesting to see if/how they act once they're state-side again ... lately, they've tried being low-key here.