To marinate the fish: Microwave the limes in a small bowl to release the essential oils. Set limes aside until just cool enough to handle. Through the feed opening of a running blender, add 1 at a time, garlic, ginger, vinegar, soy sauce, and scallions. Then, leaving the blender running, squeeze in the juice from the microwaved limes. Place the swordfish in a nonreactive container and pour the marinade over. Cover and refrigerate for 6 to 8 hours.
To marinate the fiddleheads: Place the fiddleheads in a nonreactive bowl. Make the vinaigrette, by adding, 1 at a time, through the feed opening of a running blender the vinegar, garlic, and Dijon mustard. Then, leaving the blender running, add the olive oil in a slow thin stream. Season with salt and pepper, to taste, and pour over the ferns. Cover and let sit in the refrigerator for about 6 hours. (Marinating them this way is an alternative to cooking them.)
Make the salsa while you are heating the grill for the fish. Whisk together the soy sauce and balsamic vinegar in a bowl and add the olive oil in a slow thin stream while continuing to whisk. Chop the basil leaves and add to the bowl along with the tomatoes, shallots, and scallions. Season, to taste, with salt and pepper. (Be sure to taste before seasoning, because the soy is likely to lend saltiness.)
Brush the marinated fish with oil and season with salt and pepper. Grill the fish, leaving each side undisturbed for the first few minutes to prevent tearing of the flesh. Let rest, then transfer to a cutting board and carve into slices.
Place some salsa in the middle of a serving dish, top with sliced fish and garnish with marinated fiddleheads.