A tag fight is a cardfight between 4 players, 2 against 2. Each player would have their own decks of 50 cards. There are 2 types of tag fights. One of them, you're allowed to show your hands to your team mate, and have discussions with them, where you can't on the other type. (The official website recommends allowing discussions)
Like normal cardfights, each player starts with their initial 5 cards hand and a starter vanguard. Also, each player gets to mulligan once. The winning conditions for this game format is when the opponent's team reaches 9 damage or, one of the opponent's team decks out (0 cards left in their deck).
Now to the more detailed part of the rules. Each player have their own field however, their damage is the total damage of the team. When a Heal Trigger is checked, you check damage for the total damage for your team, and the opposing team. If your team have more than or equal damage to the opposing team, you can heal 1 of your own damage. This means that you are not able to heal a team mate's damage. For Limit Break abilities, the ability is active when the total damage your team have the amount of damage required.
You can only attack cards of the player in front of you. Your team mate can help you defend against opposing attacks by placing units on their Guardian Circle however, intercepting for your team mate is not allowed. Also, you are only able to place units with grade less than or equal to your Vanguard's grade.
For paying costs, you can use your team mate's damage for Counter Blasts and your team mate's soul for Soul Blasts. You however, cannot use cards in your team mate's hand for paying any kind of costs.
To start the game, each player will play a mini-game ie: Scisors, Paper, Stone. The winner of the mini-game will go first, and then to the opposing team member which is not in front of the first player. Players up to the third player will be unable to attack, the fourth player will be the first player to be able to attack. Here's a little example if Kamui goes first. (Follow the arrows)
That's all for the rulings. Overall, I think this format is very interesting. I've tried it a few times myself and will say that this game format is a lot more fun that a normal cardfights.