Microwave lemon and lime in a small bowl to release essential oils and set aside until they are cool enough to handle.
Add soy sauce and vinegar to blender, replace lid and turn on blender. Through the feed opening add, 1 at a time, the garlic, ground cloves, rosemary, thyme, and black pepper. Halve lemon and lime and leaving the blender running, squeeze juice from each through the feed opening.
Place flank steak in a nonreactive container, add marinade and coat steak. Pour barbecue sauce over and spread to coat. Cover and refrigerate for 2 hours.
Heat grill and sear each side of steak. Cook to your desired level of doneness and transfer to a cutting board to let rest for 5 minutes.
Heat grapeseed oil in a skillet over medium-high heat. Add onion and saute until translucent, then add mushrooms and saute until they begin to give up their juices. Add red wine and allow most of it to evaporate. In a small bowl, whisk together water and beef base and add to pan. Allow to reduce by half. Season with salt and pepper to taste, remove from heat and keep warm.
Slice steak on the bias into 1/4-inch thick strips and arrange on plate. Spoon mushroom onion sauce over.