Tatishvili returned to competition at the French Open last month following a 31-month absence but lost her first-round match in just 55 minutes.
Despite winning his 12th French Open, Nadal remains in second while Barty's triumph helped her climb to No.2 behind Osaka.
Thiem beat the world number one 6-2, 3-6, 7-5, 5-7, 7-5 to set up a repeat of last year's final against Rafael Nadal.
Roland Garros organisers decided to move both women's semi-finals away from the main area and both games started at 11 a.m.
Wednesday was the second time since 2016 that all Roland Garros play was cancelled and third since 2000.
It is the second time since 2016 that all French Open play was cancelled and third since 2000.
The Spaniard has only been defeated twice at Roland Garros in 93 matches since his 2005 debut.
The pair have not played each other at Roland Garros since Nadal's victory in the final eight years ago, his fifth from five French Open matches against Federer.
Austria's Thiem, the fourth seed in the men's event and the 2018 runner-up, was conducting his post-match press obligations after reaching the last 16 late on Saturday.
Nadal swept past Argentina's Juan Ignacio Londero 6-2, 6-3, 6-3 to progress to his 38th Grand Slam quarter-final appearance.
Japanese top seed Osaka, the reigning US and Australian Open champion, crashed to a 6-4, 6-2 defeat to the 42nd-ranked Katerina Siniakova.
The sixth seed led by two sets overnight, with the match suspended at 5-5 in the third due to darkness at 9:52 p.m. local time, and saw Serbia's Krajinovic launch a comeback.
The 26th seed defeated Slovakia's Viktoria Kuzmova 6-2, 6-1 to set-up a clash on Sunday with 23rd-seeded Donna Vekic of Croatia for a quarter-final spot.
Nadal maintained his record of never having lost in the first week in Paris, although he withdrew injured before the third round in 2016.
There's big money in on-court tennis fashion and the four Grand Slam events are targeted battlegrounds.