I did not read all of the replies, so apologies if I am repeating anything.
Orvus soap. Comes in a large container, its used as carpet cleaner but is safe to use on animals and will get even the toughest stain out. Use with warm water and a stiff body brush, add elbow grease and rinse out.
Pre-washing, really give your horse a good curry. You’d be surprised how much of the stains will come out just from that.
Invest in dry shampoo spray, to use for quick-fix or on small stains. Apply amply and rub in circular motion for maximum poop stain removal.
But quite honestly, you’re going to have to live with having a not-perfectly-clean horse most of the time. Washing them too often can make them sore and cause other probles, so unless its a clinic, show, or other special event, don’t worry about those poo stains!
Eta: Keep her blanketed when you can (scrim/fly sheet in the summer) and you can also get those lycra hoods that will keep her head and neck clean, too. Again, probably don’t want to use these things all the time, but can definitely come in handy when going to a show to keep most of the filth at bay (and braids neat and shavings-free!)