Nadal (world number 5) needed five sets to get rid of a combative Denis Shapovalov (14th) 6-3, 6-4, 4-6, 3-6 and 6-3 in 4 hours and 8 minutes, while Barty walked past the American Jessica Pegula (21st) 6-2 and 6-0 in an hour.
Nadal will play on Friday in the semifinals against the Italian Matteo Berrettini (7th), who also needed five sets to beat Frenchman Gael Monfils (20th), 6-4, 6-4, 3-6, 3-6, 6-2.
Berrettini and Nadal already met in 2019 in the semifinals of a Grand Slam, in the United States. Then the Italian also won in the quarters in five sets against Monfils and then fell in three sets against Nadal.
In the women’s draw, Barty, world number one and local idol, continued to show his strength and swept Pegula off the court with that 6-2 and 6-0.
Barty will play against the American Madison Keys, number 51 in the world, who defeated the Czech Barbora Krejcikova (4th), champion of Roland Garros in 2021, 6-3 and 6-2.