Anthony Bourdain Everything Seemed Good with GF ... Until This Week

9:00 AM PT -- Asia just released a statement addressing Bourdain's passing, saying ... "Anthony gave all of himself in everything that he did. His brilliant, fearless spirit touched and inspired so many, and his generosity knew no bounds. He was my love, my rock, my protector."

She continues, "I am beyond devastated. My thoughts are with his family. I would ask that you respect their privacy and mine."

Anthony Bourdain and Asia Argento seemed as tight as ever just last week ... but it appears something changed within the last few days.

Asia, an Italian actress and director, also worked on Bourdain's show, 'Parts Unknown.' Bourdain posted a photo last week of the two from their trip to Hong Kong ... that episode just aired this week. The pic featured the crew, including Asia, who was sitting right next to him.

Just a few weeks ago, they were together in Florence, Italy ... and still looked very happy and romantic.

Here's where things get murky. We know Anthony was shooting his show in France this week -- he'd been there for at least 4 days. However, Asia was back in Rome, strolling around with a French reporter named Hugo Clément. There were photos of them holding hands and hugging, but the Italian photographer who shot the pics pulled them off the market on the heels of Anthony's death.

It's unclear if Anthony and Asia had broken up. If they did, there was no public announcement. Their last public appearance together was at an event was back in April in NYC.

Anthony and Asia started dating in early 2017, and he supported her when she went public with rape allegations against Harvey Weinstein -- saying, "I am proud and honored to know you. You just did the hardest thing in the world."