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Describe the Bug:
When a Doomwake Giant enchanted by Gift of Immortality dies it immediately comes back into play triggering its constellation effect. The Gift of Immortality should return at beginning of end step also triggering the constellation ability of Doomwake Giant, and both effects should go away during cleanup step. But the Gift of Immortality triggers the constellation ability at beginning of the next turn (don't know if this is when it returns to the bf).

Which card did behave improperly ?
Probably Gift of Immortality, but maybe Doomwake Giant

Which update are you using?(date,name)Which type(Duel,Gauntlet,Sealed Deck)
Manalink 2015/3/25: Fate Reforged,duel

What exactly should be the correct behavior/interaction ?
See above

Are any other cards possibly affected by this bug ?
depends on the cause

I confirm that Doomwake Giant doesn't trigger when Gift of Immortality returns to battlefield but the timing seems to be correct,I used Mungha Wurm ability and the Gift of Immortality trigger resolved before.

I don't understand what your test would prove. When Mungha Wurm returns to bf it becomes a continuous effect, but when Gift of Immortality returns to bf there is no trigger, so there is no timing issue.

When Doomwake Giant is the enchanted creature there are triggers which reveal timing.

What should happen:
1) after Doomwake Giant is killed it returns to bf and opp's crits get -1/-1.
2) at end step Gift of Immortality returns and triggers Doomwake Giant so that opp's crits get another -1/-1 (total of -2/-2)
3) cleanup step and effects go away
4) next turn and no effects present

What actually happens:
1) after Doomwake Giant is killed it comes back and opp's crits get -1/-1
2) end step and effects go away
3) next turn and Gift of Immortality triggers Doomwake Giant and so opp's crits get -1/-1 for that next turn

If you replace Doomwake Giant with Eidolon of Blossoms then the card drawn for Gift of Immortality returning isn't drawn until the next turn. So I think it is Gift of Immortality returning at beginning of the next turn instead of at the end of the current turn.

This clarifies that there is bad timing but it is the clean up phase when Gift of Immortality returns instead of the beginning of next end step that is why Doomwake Giant effect goes until next players end step and every other damage-effect has cleared up.

The probelm is that "PHASE_CLEANUP", in which the legacy card for P/T modification are eliminated, happens BEFORE EOT trigger in Manalink, so you wouldn't never get the -2/-2 bonus, as the second -1/-1 will be created after PHASE_CLEANUP. I don't know if this is solveable.

Moving the phases around is easy - neither cleanup_phase() nor end_turn_phase() are at all complex, and even if they were, their call order could be swapped. The real problem is all of the cards relying on the current order.