Ah. Sorry to be the bearer of bad news, but I don't think it's diatom algae. Doesn't look like it, and diatom algae brushes off easily. Very soft, a lighter brown, and can form a sort of carpet layer even, but is easily sucked up with a gravel vac and washed off of fake plants.
This stuff, let me guess, is really hard to remove, even if you take a fake plant out and scrub at it?
I don't know the name of it, but it looks very much like a nightmare algae that I helped try to eradicate from my dad's tank, that did have live plants, but only cryptocorynes. It stuck to everything so hard, it was almost impossible to remove. Tried several times to remove every leaf that had it, remove any decor that had it and scrub it off, but it always came back with a vengeance. I'm afraid to say that it didn't go away until we did a complete tank tear down.

Cryptocorynes cut right down until only roots left, substrate removed and cleaned, and lots of uncleanable decor thrown away and replaced.
If it's the same thing, then I'd do the following:
Fake plants you can remove and try a bleach clean and scrubbing them clean. You'd need to look up the dilution recommended for bleaching tank ornaments since I don't know it offhand, but it can be used with live plants, so would be fine for fake I'd imagine. After scrubbing as much off as you can and doing a bleach soak, to be safe, rinse really well and then soak the plant in a triple dose of declorinator, then rinse some more and allow to air dry completely before replacing in the tank. If it's not all gone, I'd throw them and replace.
Wood and stone is a different matter and can't be bleached but can be scrubbed as much as you can and baked in a hot oven to try to kill it off.
Scrape down the inside of the tank walls with a Stanley blade, to remove any traces that might be clinging to the glass, right before doing a large water change/gravel vac to remove as much of the free-floating yuk as possible.
Replace clean decor and hope that it worked!
Then I'd add a bunch of live plants. Both because I'm a planted tank nut, and think every tank looks incomplete without some, but also because some fast-growing live plants (the crypts in our case are pretty slow growing so not as helpful) can outcompete the algae for nutrients, if the balance is right. Make sure you're not overfeeding. Some floating plants can also provide some shade. This algae seems to thrive when it has more lighting.