You cannot venture into Undercity unless instructed to do so, either because you have the initiative at the beginning of your upkeep or because you take the initiative. Notably, if you aren’t in a dungeon and an effect instructs you to venture into the dungeon (not venture into Undercity), you can’t start Undercity.
Tomb of Horrors Adventurer
Similarly, when instructed to venture into Undercity, you can’t start a dungeon that isn’t Undercity.
In a Two-Headed Giant game, if both players on a team deal combat damage to the player that has the initiative at the same time, the player with the initiative will choose the order of the triggered abilities. Then, as those abilities resolve, one team member takes the initiative (and ventures into Undercity) and then the other team member does the same. The last player to take the initiative keeps it until the initiative changes again.
Magic: The Gathering · ™ & © Wizards of the Coast · Illus. Irina Nordsol
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