Since most wrestlers theoretically have little say on who they either fight or team with, an Enemy Mine often occurs when an on-screen authority figure randomly decides two people who don't like each other will team up against a common enemy. This used to happen all the time with "Stone Cold" Steve Austin and The Rock. An example would be when Stone Cold was booked to fight The Undertaker and Rock against Triple H at Over the Edge 1999, one Raw saw Austin and Rock being forced to team together against Undertaker and Triple H. Since Undertaker and Triple H were members of the same stable, the story of the match was whether Austin and Rock (who only weeks earlier were in a bitter feud) could put up with each other long enough to survive.
This specific Enemy Mine would occur several times through 1999, not all of them being in official matches. For example: the SmackDown pilot in which the CorporateMinistry came together in the first place.
Stone Cold and The Rock would continue to find themselves unwilling partners as late as 2002, when they teamed up the week before WrestleMania X8 to face the New World Order in a handicap match.
A Pro Wrestling tag-team championship can also serve as an Enemy Mine, as a face and a heel are thrust into a match with the titles on the line, and despite their bickering and contempt for each other, manage to win. They're then obligated to continue teaming to defend the championship, arguing and taking potshots at each other all the while, until one of them finally decides being a champion isn't worth the headache and turns on the other. A 2007 example is Matt Hardy and M.V.P. on WWE SmackDown!.
Slightly subverted in 2008 with John Cena and Batista, who won the tag titles but were bitter enemies facing each other, and couldn't co-exist, and lost the tag titles back on their first defense the next week.
Even more so with Cena and The Miz who couldn't co-exist as tag team champions for ten minutes before Miz betrayed Cena.
As an interesting side note, all four of John Cena's tag title reigns in WWE have been with enemies or rivals, and each one was shorter than the last as a result. His first World Tag Team Championship reign with Shawn Michaels came during the lead-up to their Wrestlemania match and ended the night after that event, bringing their total reign to 63 days. His second World Tag Team Championship reign was the aforementioned reign with Batista, which lasted seven days. He later won the WWE Tag Team Championship with David Otunga as an unwilling member of The Nexus, losing the title the very next day. His final WWE Tag Team Championship reign was the one with The Miz mentioned above, lasting about ten minutes.
Hilariously lampshaded in a backstage segment with Team Triple H at Survivor Series 2007. Kane, Jeff Hardy, and Rey Mysterio call out Triple H on his numerous past offenses against them, which "the Game" sheepishly apologizes for. However, the team acknowledges that while they don't necessarily like one another, they like the other team even less.
Subverted at SummerSlam 2010, when usual Cena rival The Miz tried to play on the "need" for this (especially after Cena honestly tried to keep them together, but Edge and Jericho proved incompatible) to make Cena beg him to round out the team... by the time Miz tried to pull the trigger, Cena declared that Miz had taken too long to make up his mind and that Cena had found someone else who said yes first: Cue the triumphant return of one Daniel Bryan.
Team Hell No is a great example, with Daniel Bryan and Kane being forced to enter anger management together, hug out their aggression in the middle of the ring, and become a tag team. They won the tag team titles completely on accident, and they both would spend most of their time bickering and arguing, which through sheer luck would lead them to win after win.
In a interesting twist, The Rise and Fall of ECW showed that the Tazmaniac and Sabu hated each other in real life, but ended up becoming one of the company's top tag teams prior to Sabu bailing on the company before The Three Way Dance on April 8, 1995.
The American Wrestling Association had an angle in 1985 where the Russian faction of Boris Zhukov and Chris Markoff injured Rick Martel. Martel's friend Mad Dog Vachon came looking for revenge and turned to the only man he felt was dirty and ruthless enough to stop the Russians: his old enemy Nick Bockwinkel.
Often happens in Royal Rumbles and Battle Royals, as many wrestlers team up to eliminate the biggest wrestlers.