As counter-intuitive as it might sound to observers of American politics, democratic might work as a synonym.
democratic: Relating to or supporting democracy or its principles
Republican is even listed among the possible synonyms of democratic, although the list contains less fitting words. (Link) Since republics (should) have a government which uses elected officials representing the citizen body (Wikipedia) this is not that surprising.
However, it is a linguistic synonym, which might seem strange in America with the Republicans and Democrats being as partisan opposites as they are today.
EDIT: Regarding democracies and republics
Today the term republic usually refers to a representative democracy with an elected head of state, such as a president, who serves for a limited term; in contrast to states with a hereditary monarch as a head of state, even if these states also are representative democracies, with an elected or appointed head of government such as a prime minister.(Wikipedia)
However, these constitutional monarchies are sometimes dubbed crowned republics:
A crowned republic may thus be a form of parliamentary republic, while officially it may be called a constitutional monarchy.
Bonus (same Wikipedia article):
The novelist and essayist H. G. Wells used the term to describe the United Kingdom, as did Alfred, Lord Tennyson in his poem Idylls of the King.