How do I stop my Liane 360 from pulling away from the ceiling where it curves?

I ran into the same problem on mine. The corner section is under constant pull in two directions at once, so adhesive backs and even the sticky clips just peel eventually. Toggle bolts are the right call for a ceiling install like that, honestly. Drywall anchors alone will hold for a while but ceiling loads work against them differently than wall loads, and the constant sideways tension at a bend makes it worse. A couple of toggles with small clips or even zip ties looped thru will hold that bend permanently. I did toggle bolts at my corner and it hasn't moved in over a year. Just be careful when you're up on the ladder drilling overhead, and hold a rag under the hole beacuse drywall dust falls straight into your eyes otherwise.

On the bend tightness, the strip can handle a corner but you don't want it kinked. If your curve looks sharp, try to spread it out so the transition happens over a longer run instead of a tight elbow. What helped me was treating the bend as two flat sections that meet at a gentler angle rather than one continuous swoop. Less tension means less pull on whatever mounting you use, and it also keeps the light diffuser from creasing, wich shows up as a wierd bright line when the colors run.

One other thing, when I re-did my corner I actually repositioned it slightly so the bend landed where a ceiling joist was. If you can hit one with a regular wood screw instead of an anchor, that's the strongest option of all. A stud finder makes finding them quick. Between that and the toggles for the rest of the run, mine is solid.

Mine did the same peeling thing at a bend and what finally fixed it was a tiny dab of clear silicone behind each clip before screwing them down. Cheap and it hasn't budged since. Also, if you're re-sticking anything, warm the adhesive with a hairdryer first, it grabs way better than cold.

Out of curiosity, how'd you get the swoop looking that clean? Every time I've tried a curve like that mine ends up with flat spots where it wants to kink. Did you ease it in gradually or just commit and live with a sharper turn?

One thing I'd add is to check which way the strip is actually bending at that transition. Mine wanted to twist slightly edge-wise instead of folding flat against the ceiling, and that twist was doing more damage than the curve itself. I put a small right-angle bracket at the corner so the two legs of the run are each supported on thier own plane, and the tension basially disappeared. Also, if you ever have to re-seat the strip, press it on and leave the lights off for a few hours so the backing can set properly before it's working against a load.