Savory Blackberry Sauce
I'm sure all the other recipes work out great.
This is what I do when I want a tasty berry sauce.
Mix two healthy tablespoons of jam, a pinch of kosher salt, a pinch of chipotle or cayenne pepper, and a tablespoon of stock (chicken or vegetable) in a small bowl. Immediately after you've taken your loin off to rest, put your mixture into what you've cooked it in (with all the juice and oil), and stir gently until it starts to bubble and thicken. Pour over you meat and your good to go. If you want to make it even less sweet, add lemon juice to the mixture.
You may also enjoy making a gastrique, which can be made in advance and stored in the fridge for several days.
Basically, 1 cup honey, 1 cup vinegar such as sherry or banyuls, plus 1 cup berries. Simmer on low until reduced by 2/3. Strain through fine mesh sieve. It will be thick and will coat your meat perfectly. This is a version based on Thomas Keller's cherry gatrique.
Here's another, to give you an idea. They sound fancy, but are so easy it's ridiculous!
In a heavy bottom pan, brown your seasoned pork in evoo. Drain all but 1 tbsp. of fat. Tent the pork and/or place in warm oven.
Saute sliced onions and thyme in renderings and start to get the wonderful fond up.
Deglaze with port, about 1/3 cup and reduce a bit. Get the rest of that fond up
Add blackberries, blackberry jam (or actually apricot jam works well too!) and chicken stock.
Simmer until thickened such that the pan appears when you run a wooden spoon across the bottom.
Add back any juices from the pork.
Plate pork with thickened reduced sauce.
I've been using the book Berry Bible for black currants. It has a recipe for blackberries that it uses with lamb that might work. Here's what it has:
1.5 cups port or madeira
2 tbsp honey (optional)
1tbsp fresh lemon juice
1/2 tsp fresh thyme leaves
1.5 cups blackberries
The recipe has you mix the first ingredients except the blackberries together and then pour over the meat. Add the blackberries. Cover and bake or bake in crockpot until done (quite a while if using the lamb shanks.) and the meat is tender. Discard accumulated fat, season the juices with salt and pepper and serve the sauce over the meat with some fresh berries for garnish.
Another option might be a black currant sauce adapted for blackberries that the book has you serve with duck breasts. That recipe has shallots, sherry vinegar, duck or veal demi-glace (skip?), chicken or meat stock, a little dijon and butter adn a few tablespoons of black currant syrup. That syrup has sugar in it but I think it would get modified by all the other things like the sherry vinegar and stock. Let me know if you want more info on that one.