What is the most number of pitches thrown by both teams in one inning of a Major League Baseball game?

This might be it, as it was the longest-ever MLB inning. There were 112 pitches in the 5th inning in the Detroit Tigers at Texas Rangers game on 9th May 2004.

The longest MLB inning occurred on May 8, 2004, when the Detroit Tigers and Texas Rangers combined for 18 runs in a 5th inning that lasted 68 minutes.

https://sports.yahoo.com/article/longest-baseball-game-ever-mlb-220137799.html

This webpage says there were 100 pitches in the 5th:

The Texas Rangers and the Detroit Tigers combined for 18 runs in the fifth inning -- one short of the major league record for an inning -- and Alfonso Soriano went six for six in the Rangers' 16-15, 10-inning victory Saturday night in Arlington, Texas. Michael Young hit an RBI single off Ugueth Urbina (1-1) in the 10th to win it for the Rangers, who won after trailing by 10 runs -- the first time in club history they accomplished that feat.

Each team used seven pitchers in the game, with Francisco Cordero (1-0) pitching a scoreless 10th. Eric Young walked leading off the bottom half, was sacrificed to second and scored on Young's hit after an intentional walk to Brad Fullmer. After Detroit took a 14-4 lead with eight runs in the top of the fifth, the Rangers tied it in the bottom half. Ivan Rodriguez put the Tigers back ahead with an RBI single in the sixth, but Hank Blalock tripled leading off the seventh and scored on a double by Soriano, who raised his average from .302 to .336. There were 100 pitches in the fifth, which lasted 1 hour 8 minutes and became the highest-scoring fifth inning in major league history. The bottom half lasted 36 minutes.

https://www.latimes.com/archives/la-xpm-2004-may-09-sp-al9-story.html

However, the box score at baseball reference.com shows 54 pitches top of the 5th and 58 pitches bottom of the 5th (total 112), if I'm reading Play by Pay correctly:
https://www.baseball-reference.com/boxes/TEX/TEX200405080.shtml