When does the EXALT faction appear in XCOM: Enemy Within? Is it random or after a specific "story point," research milestone, or council mission?
EXALT always starts its operations at a random date in May, with the possible variation between dates dependent on the game's difficulty level. On Easy/Normal difficulties the day can vary between the 12th and the 24th, while on Classic/Impossible it can start anywhere between the 6th and the 24th.
For reference, the game begins in March, so May is the third in-game month.