Combine orange juice, 2 tablespoons cilantro, lime juice, garlic, salt, pepper, cumin, and cayenne in a large freezer bag. Submerge skirt steaks in the marinade. Seal the bag, squeezing out as much air as possible. Refrigerate for 1 to 8 hours.

Preheat a grill to medium-high. Grill the steaks until hot and firm in the center or to your preferred doneness, approximately 6 to 8 minutes per side. Transfer steaks to a flat work surface and slice as thinly as possible against the grain.

Heat corn tortillas on the grill or in a microwave. Keep tortillas warm in a folded sheet of aluminum foil covered by a thick towel.