To prepare marinade, combine preserves, hoisin, lime juice, cilantro, ginger, and sesame oil in a small bowl. Stir well.
Pour into a resealable plastic bag or glass bowl; toss tuna with marinade to coat. Refrigerate for 1 hour.
Preheat an outdoor grill for high heat and lightly oil grate.
Remove tuna steaks from marinade, shake off excess, and discard remaining marinade. Cook on preheated grill for 4 to 5 minutes per side. Be careful not to overcook - the tuna should be lightly browned on the outside but still slightly pink in the middle.
