An example of where this is not allowed is Seattle, WA. Municipal code SMC 12A.06.025 states
It is unlawful for any person to intentionally fight with another
person in a public place and thereby create a substantial risk of:
Injury to a person who is not actively participating in the fight; or
Damage to the property of a person who is not actively participating in the fight.
B. In any prosecution under subsection A of this Section 12A.06.025,
it is an affirmative defense that:
The fight was duly licensed or authorized by law; or
The person was acting in self-defense.
You can see from adjacent sections that "mutual combat" is not legal. I recognize that there is this meme about Seattle, but this is a distortion of an incident when the police turned a blind eye to a fight. We have police issues, no doubt: there is nothing legal about such fights. Of course, for a licensed event, you can "fight".
Of course the potential legality depends on how mutual combat is defined. Illinois v. Austin 133 Ill.2d 118 and citations therein, subsequently Illinois v. Thompson, 821 NE 2d 664 define it thus:
Mutual combat is a fight or struggle which both parties enter
willingly or where two persons, upon a sudden quarrel and in hot
blood, mutually fight upon equal terms and where death results from
Similar death-definitions are found in Donaldson v. State, 289 SE 2d 242, Iowa v. Spates, 779 NW 2d 770. The law looks askance of such behavior. For the sake of clarity, a term other than "mutual combat" would be preferable.