Officers said about 40 protesters gathered at Bristol Cenotaph aiming to march through Castle Park and back to the same point.
Police said they had to redirect them to keep the two groups separate.
Police also arrested a 39-year-old man on suspicion of obstructing or resisting a constable and a 23-year-old man on suspicion of violent disorder.
A woman, 23, was arrested on suspicion of assaulting an emergency worker and obstructing or resisting a constable in execution of their duty.
Another woman, 27, was arrested on suspicion of failing to provide details upon request and acting in an anti-social manner.
Officers from Wiltshire Police, Gloucestershire Constabulary, and Devon & Cornwall Police came to assist, with horse and dogs units deployed.
By 15:30 GMT, police said attendees had dispersed.
