Hello Besthost, welcome to Using English!
When two contestants in a competition have an equal number of points, it's called a "tie".
To decide which contestant is the winner, the question-master will ask a special question, called a "tie-breaker".
The person who answers the "tie-breaker" correctly wins the contest.
In your context, "campaign strategies" can have the effect of a "tie-breaker", where two candidates are equally matched – in other words, they decide who wins.
"They can take a tie and break it" isn't a set phrase: the speaker has simply extracted the notion that is implicit in "tie-breaker".
All the best,
Student or Learner