Let's play a game of make believe:
Imagine you get a call from your ex wife. She tells you that your 15-year old daughter, Natalie, has been invited to a party/sleepover at another friend's house. The party is on a weekend that Natalie will be with you. Both the party and the sleepover will be co-ed. Your ex wife doesn't have any problem with the concept of a co-ed party; but she strongly objects to the co-ed sleepover part. She explains to you that she's already told Natalie as much, and that Natalie is now furious with her. Natalie thinks she's being annoying and lame, and insists that "everybody" is going. Your ex wife asks for your support. She would like for the two of you to present a united front to Natalie by explaining to her that neither of you think that a coed sleepover is okay.
You listen quietly, the wheels in your head turning. You think to yourself, 'This is one of those perfect opportunities! I don't like the idea of coed sleepovers any more than my ex does, but I do like the idea of being popular with Natalie. Since her mom has already gone on the record with the "lame and annoying" decision, I can come out of this looking like the cooler, more sympathetic parent. I'll tell Natalie that since her mother has already made a decision, there's nothing I can do about it. That way, I am not opposing my ex, so technically I'm good there. But more importantly, Natalie and I are in a position of solidarity: We both are victims of her mom's uptight, tyrannical style! I get the outcome I want (Natalie doesn't go to a coed sleepover), I don't get any flack from my ex (after all, I didn't contradict her decision), my ex takes all the bullets (the decision is pinned on her), and Natalie and I become closer (I "get" how impossible her mother is). The wedge between Natalie and her mom grows just a little bigger with each round of this game. It's a win/win/win/win/win situation. I. Am. BRILLIANT!'
The plan you just crafted is neither original nor smart. In fact, this approach is so common that it already has its own entry in the Lexicon of Divorced Co-parenting. You'll find it under "spineless support." Spineless support is when a parent deliberately leaves the tough decisions and unpopular positions to his ex, so that he can be more popular with his kid by comparison.
Those who employ this strategy usually know at some level that it's unfair to their ex because it forces her into the role of bad cop; but they typically fail to realize that the strategy harms their kid by increasing friction between her and her mom. Unfortunately, those who give spineless support aren't mature enough to see that as a negative for the kid. (If they were that mature, they wouldn't employ this strategy to begin with.) But this isn't the only way it harms the kid. It also costs her the opportunity to gain a clear understanding of the principles for which both parents stand, and it dilutes the parenting power that she has in her corner because one parent is more focused on popularity than parenting.
If you find yourself tempted to give spineless support, there is something you can do. Before you open your mouth, take a minute to gather your strength. Don't give an answer until you have summoned the courage to give your kid what she really needs: your full and genuine support. Rather than tell her that her mom has made a decision, or that it's up to her mom to handle it, or that she has to take it up with her mom, have the spine to tell her what you actually think about the issue.
It may cost you some popularity points in the short run, but if you stick with it, you and your kid will be richly rewarded. You will have a stronger, healthier relationship with your kid; she will know she can really count on you because you care about what's best for her, not what's best for your popularity with her; and all of that will increase her chances of steering clear of trouble and growing into a well-grounded, healthy individual.
Co-parenting with an ex is hard work. Parenting a teenager is likewise daunting. Doing both of these things at the same time requires all the resources at your disposal. So, use your spine and give your kid the full strength support that she deserves.