When we are not playing competitive games, our three touring teams play against three exhibition teams.
What do the bold terms mean?
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An "exhibition team" is one which plays "exhibition games", or a game that has no competitive value, i.e. rankings, prize money like a friendly match. Wikipedia defines this:
An exhibition game (also known as an exhibition match, exhibition, demonstration, demo, exhibit or friendly) is a sporting event in which there is no competitive value of any significant kind to any competitor (such as tournament or season rankings, or prize money) regardless of the outcome of the competition.
A "touring team" is one which travels around playing other teams to compete to see who was better, i.e. In the world cup, the touring teams of the countries all over the world gather to compete.
Thus, your above example meant that when the three teams that usually traveled around competing with other teams were not doing so, they just played friendly matches.