1 cat attacks the other

Here is the situation - 2 cats (Ginger 2 y/o and Annie 18 m/o) were adopted from a no-kill shelter. Both cats came from the same room with 10 other cats at the shelter and after a couple of days of adjustment both have lived in harmony. Indoor with plenty of room in 2 story house, 2 separate litter boxes, 2 separate window perches, both are spayed.
About 6 weeks ago Ginger started attacking the Annie. Annie is the non-aggressor, runs, Incontinence when attacked, just hisses and defends her self. The attacks involve Ginger chasing Annie, growling, screeching, very little blood, maybe one scratched nose.
We took both to the vet and the vet gives them a clean bill of health. Tells me to keep them separated and re-introduce them later on. We think Ginger saw another cat outside the window and flipped out on Annie. We have kept them separated, rotating them out the rooms, etc. They both know the other cat is in the house.
Last night, I was moving Ginger into my daughterâ€[emoji]8482[/emoji]s room for the night, she was on the bed, I opened the door and she saw Annie in the hall, launched herself off the bed and out the door chasing Annie downstairs.
We have tried feliway upstairs and downstairs with no luck. Separately both cats are a joy to be around.
My wife has about had it and ready to return the aggressor back to the no-kill shelter.
Any ideas?

Some cats just need to be only cats.

I sympathize with you on this one. I brought in a feral older kitten, and she has terrorized my two neutered male cats. So I isolated her once again, got her spayed, and will slowly reintroduce. I, too, have purchased Feliway - it's been plugged in a week, but I don't know if its working or not yet.

Years ago when I still lived at home we had a friendly indoor/outdoor neutered male cat who could hunt like the devil, but was a wimp with other cats. We adopted an abandoned female and she spent her days chasing him down the hall and under a bed. He was terrified of her. I remember thinking if he would just one time stop and face her, it would probably end. Well, in time, it did end. They both lived to be 20, and co-existed well. Unfortunately, neither my mother nor I can remember what happened to change the situation for the better. It apparently just happened.

Keep the faith. Maybe some cats are meant to be only cats (as I'm sadly afraid my formerly feral is), but I think if you give it enough time they'll learn to exist together without fighting.

My younger cat did this to my older cat when I first got him. Clive was 10 and a kitten was probably not the best idea. Didn't realize...
As he got older Duke stopped, but I did have to break them up a lot.
Try squirting with a water bottle when they are going at it.