Kate Middleton Seen in New Video Enjoying Windsor Farm Shop with William

The Kate Middleton sighting that got reported on this week is real after all -- 'cause a new video of her during this outing has surfaced ... and TMZ's got a first look.

TMZ has obtained a clip that was taken of Kate Saturday while she and Prince William were visiting a business about a mile from their home in Windsor -- known as the Windsor Farm Shop -- where The Sun reported eyewitnesses saw them out and about this past weekend.

Onlookers felt she looked "happy, relaxed and healthy" as she strolled in the store -- a supermarket of sorts -- and the couple didn't seem to have a care in the world.

Their three kids weren't with them ... but The Sun also claimed they spent the first part of their Saturday watching the children play sports. At first, there was no real evidence to back up the report ... but now, the first actual visual of her here has surfaced, and it's 100% her.

As you see in this footage, obtained by TMZ, Kate looks relaxed in casual athletic wear ... rocking an all-black look with some pink trimming. Her gait is strong -- a good sign given her purported abdominal surgery. PW is right by her side as they walk out of the Farm Shop with bags in hand.

Nobody around looks to be bothering them ... and Kate, sure enough, does indeed look pretty cheerful here -- smiling wide and even carrying on a conversation with her husband.

One of the eyewitnesses who saw them, Nelson Silva, says ... "I noticed a couple choosing loaves of bread and the woman turned her face and I felt like I had seen the face before. It was familiar. I knew it from somewhere."

He adds, "I went to my car and as they came out of the shop I just filmed them. I think they walked out through a gate out of the grounds. They just vanished and I didn't see a car. I just wanted to share with my family footage and show just how normal they were."

Silva finishes ... "Kate looked happy and relaxed. They look happy just to be able to go to a shop and mingle. Kate looked relieved like it was a success going to a shop. It felt natural."

The couple was walking at a relatively brisk pace ... and Kate, on its face, seems to be okay. Frankly, it's as if nothing was ever wrong with her ... although, of course, we know she was in the hospital for a while for a serious procedure that took her out of commission.

The exact nature of her ailment was never made clear by Kensington Palace officials.

BTW, for all the skeptics who might not believe this was taken as it's been relayed to us -- we've delved into the metadata ... and there's no doubt, this was filmed on Saturday right near the Prince and Princess of Wales' home in Windsor. So yes, it's a legitimate video.

Like we said, this is the first time we've seen Kate out in public -- outside of a car, that is -- since December -- and now that we're seeing her in the wild, on her own two feet, ya gotta wonder if this will satisfy conspiracy theorists who have taken over TikTok.

Remember, the Palace says she underwent abdominal surgery in January -- and was scheduled to be sidelined until at least Easter. Despite this timeline ... that hasn't stopped people from floating jokes/conspiracy theories, which got gravely serious on Mother's Day.

Of course, we're referring to the Photoshop fail seen 'round-the-world ... which showed that Kate (or someone at Kensington Palace) tried to pull a fast one on the public. There were 16 alterations made to the photo ... and it's been debunked as a fake many times over.

Kate ended up copping to it herself ... albeit, in nothing but a social media statement. Lots of people felt she was falling on the sword and that the Palace was behind the edits.

After that botched job ... everyone thought something very serious was going on -- but now, we're seeing Kate out and about and seemingly fine.

Are you believing your lying eyes?