There are many synonyms for the phrase "were rivals." Some possible options include: competed against each other, were adversaries, were opponents, were enemies, challenged one another, vied for supremacy, battled for dominance, contended for superiority, engaged in a feud, squared off, and went head-to-head. Each of these phrases connotes a sense of competition and tension between two parties, suggesting that they are striving to outdo each other or gain the upper hand. Depending on the context in which the phrase is used, different synonyms may be more or less appropriate, but all convey a similar sense of rivalry.