I'll give you some Anna and Matthew headcanons because that's what I've got! It's the primary context through which I think about Anna, lol, so.
They try out new clubs and salons every month and see which ones they like. The Hell Ruelle may be their favorite, but they have other haunts, too.
Matthew constantly lends Anna nice clothes to wear out, and she lends him some periodically, too, when they fit. She has all of her men's costumes tailored to frame her curves better, and this leads to them not always fitting Matthew's masculine frame.
They have tea and gossip about everyone in London once a week.
Anna is one of the first people Matthew came out to, and Matthew is one of the first people that Anna came out to.
Anna was always trying to warn Matthew away from certain men and women that she knew wouldn't treat him properly, but this just gave him fuel for his self-destructive tendencies. She never figured it out.
She definitely yelled at Kellington once or twice.
Every time Matthew had some passion that didn't pan out, Anna would show up at his door with a Charlotte Russe cake (canonically Matthew's favourite dessert per an old newsletter). They would each grab a fork and eat it straight from the tray, no slicing necessary.
Matthew admires Anna because of her "hardness" and seeming inability to get a broken heart. Matthew always has a broken heart. It feels like an impossibility to him. Little does he know...
Matthew helped Anna decorate her flat when she first moved in. He took the job very seriously and did all in his power to make the tiny space look aesthetically pleasing. He made sure there was a lot of natural light to bounce around the space and gave her old plush furniture from the Devil and his own house.
When Virginia Woolf gave Anna Percival, she didn't tell Matthew. Instead, she invited him over and put him in the entryway of her home. Matthew almost pissed himself and screamed bloody murder, but eventually thought it was a hilarious lark when he calmed down. He was proud of her.
Matthew would give Anna things that made her feel more masculine as a child. There's some canon basis here, what with him handing her menswear in EET. But like, if there was a party with "boy favors" and "girl favors," he'd trade with Anna. It was no problem for him - he secretly liked the frills and flowers anyway.
After ChoT, Matthew plans meticulous romantic dates for Ari and Anna. He lets Anna take all the credit, of course, and Ari swoons.
Matthew is Anna/Ari's first child's godparent.
Matthew and Anna go to Paris together every couple of years as adults. Even when they live far apart, they make it a point to meet there sometimes. Periodically, Thomas and Alastair come along, as do Ari and Matthew's partner. It's basically the Big Gay Club France Meetup. They have croissants.
Writing this out has made me remember how much I love their friendship, lol. Thank you.