The have a Rival League, you need to have an established Major Sports League operating. In the early beginnings of organized sports leagues, there were attempts to form a sustainable league but eventually they would fail…
Roster of Rival Leagues by Year
American Association (Baseball, 1882-1891) – 8 teams absorbed to NL
Union Association (Baseball)
Player’s League (Baseball)
Roster of Rival Leagues by Sport
Totals:
Baseball with 2 Rival Leagues (1882 to ?)