Combine white miso, mirin, sugar, and water in a saucepan.

Heat over medium heat, stirring until sugar dissolves.

Remove from heat and whisk in mayonnaise.

Cool the marinade to room temperature.

Place tuna cubes in a sealable bag or dish.

Pour marinade over tuna, ensuring all pieces are coated.

Marinate in the refrigerator for at least 1 hour (up to 24 hours).

Prepare grill for direct-heat cooking (high heat for gas).

Thread tuna onto skewers, leaving a small space between pieces.

Oil the grill rack generously.

Grill skewers, turning once, until tuna is just pink in the center (about 4 minutes total).

Let the skewers stand for 5 minutes before serving.