Cleveland can tie the 1935 Chicago Cubs with a win this afternoon for most consecutive wins in a season (21) against a Detroit team than has tanked. But that's another subject.
The 1916 New York Giants had a 27 game unbeaten streak: 26 wins, one tie in the middle. I'm not even sure when the most recent tie game occurred. One hundred years ago they stopped when it rained too much or got too dark. Now they start games hours late and/or have multi hour delays in order to reach a conclusion. Also, they play many extra innings.
Just accept tie games as part of the natural order. Base standings on wins or winning average (percentage).
There were ties in the 1960s. In 1961 the Yankees played a tie game, giving Roger Maris yet another game in which to break the season home run record of 60 set by Babe Ruth in 1927 for the Yankees, who also had a tie game. Ruth hit number 60 in Yankee team game 154, including the tie. Ruth did not homer in team game 155. Maris hit number 61 in team game 163.
In 1968 the Yankees played to a 19 inning tie against Detroit in the second game of a twi-night doubleheader because starting that season the old American League instituted a 1:00 AM curfew: no inning could start after that time.
No overtime in the regular season, maybe not in the tournament either.