To Roast Garlic - Cut about ½ inch off of top of garlic. Cut the roots so that the garlic will sit flat. Remove the papery skin from the garlic, but leave enough so that the cloves stay together. Put garlic into a small casserole dish or baking pan, drizzle olive oil over it, and cover with a lid or foil. Bake in a preheated 325°F oven for 1 hour. Remove garlic and let it cool until you can handle it.
Combine water, pineapple juice, teriyaki sauce, brown sugar, tomato paste, bay leaves and allspice in a medium saucepan over medium/high heat. Stir occasionally until mixture boils then reduce heat until mixture is just simmering. Simmer for 30 minutes.
Squeeze the sides of the head of garlic until the pasty roasted garlic is squeezed out. Place 12 cloves in a small bowl and gently mash. Add to the sauce and whisk to combine. Add remaining ingredients to the pan and stir. Let mixture simmer for 1 hour or until sauce has reduced by about half and is syrupy. Make sure it doesn't boil over. Adjust seasonings as necessary. Remove from heat and remove bay leaves, lemon slice and lime slices. Discard. Let cool completely before using. May be kept refrigerated in an airtight container for up to 4 weeks.