The Puritan Anthony Burgess wisely answered this question by making a careful distinction. He said that we have to distinguish between repentance taken "largely" and repentance taken "strictly."
Largely speaking, repentance includes faith. It is godly sorrow and turning from unbelief to belief, from lovelessness to love, and from disobedience to obedience. Largely speaking, repentance can even be said to include the whole of salvation, a process not completed until the last day. Largely, we may say "repent and believe."
But, strictly speaking, repentance is distinguished from faith. When repentance is precisely defined, it may not be defined to include faith, since that would be to "confuse" (join together) repentance and faith and thus fail to distinguish systematically. When speaking strictly of repentance, it always comes after faith. First, we believe in Christ. Second, and on the basis of faith in Christ, we have godly sorrow for our sin and turn from it. Strictly speaking, we say, "Believe and repent."
See Joel R. Beeke and Mark Jones, A Puritan Theology: Doctrine for Life (Grand Rapids, MI: Reformation Heritage, 2013), 330-331.