Pour root beer into a medium large pot. Bring to a boil, then reduce heat slightly to a maintain a constant simmer. Cook for 30-45 minutes or until root beer is reduced in half to about 1 liter.
Remove from heat and add barbecue sauce, garlic, ginger, sriracha, salt and 2 teaspoons fresh rosemary.
Cut pork into several smaller pieces if using a large loin. If using pork tenderloin, don't worry about cutting the meat up.
Transfer liquid to a large slow cooker. Submerge meat in liquid. It probably won't be completely covered but that's fine. Turn cooker to high and cover. Cook for 4-5 hours or until meat is tender when pierced with a fork. Pork tenderloin will take less time than pork loin, probably around 2 hours. Start checking the meat after about 3 hours.Turn meat pieces once or twice during the cooking process. Don't overcook or pork will be dry.
Remove meat from liquid and allow to cool slightly before shredding.
Transfer liquid that remains to a medium-large size pot. Skim fat from surface. Simmer for 30 minutes or until it begins to thicken and is reduced to about half the volume. When reduced, taste and add salt and pepper, if needed. Add last teaspoon of fresh rosemary.
While sauce is simmering, shred the pork with two forks in nice bite size pieces. Don't shred it too fine or it will become mushy. k.
When sauce is reduced, add about half of it to the shredded meat and gently stir to coat the pork. Again, don't stir too vigorously so that pork pieces break apart. Reserve the rest of the sauce.
Serve pork on buns or sandwich rolls of your choice. Top pork with Honey-Lime Slaw (recipe below), as desired. Pass reserved sauce at the
table.