I've pulled the most relevant sources I could find for you, though there are probably more. This is from the Adventurer's League FAQ:
Rules from an official D&D Adventurers League
source, such as the Adventurers League Player’s
Guide (ALPG), the Adventurers League Dungeon
Master’s Guide (ALDMG), or this FAQ establish the
boundaries for our current campaign.
From the ALDMG, with emphasis added by me:
As a D&D Adventurers League Dungeon Master, you
are empowered to adjudicate the rules as presented
by the official materials (PHB, DMG, MM, etc.). Run
the game according to those rules, but you are the
final arbiter of any ambiguities that might arise in
House rules, that is to say rules that you create
that aren’t in the official materials such as critical
fails, new races, new classes, etc., aren’t permitted
for use in D&D Adventurers League play; the
Adventurers League uses the rules as presented in
If a DM isn't running a game by the Adventurer's League rules, such as by allowing non-legal characters to play, they themselves aren't following the rules. That would indicate that it isn't a legal game. Bearing in mind that they're allowed to arbitrate ambiguities, though, if it's uncertain whether a character is legal, and they make the call to continue anyway, the game itself would still be legal by the mighty powers of DM fiat.
If you have any doubt about whether a character is legal, the AL Player's Guide gives you a list of allowable source material for the build and rules on how to make a legal character.