What have you found works best for building a sales culture that is competitive but also cooperative?

It requires a lot of communication and setting the expectations by the leader. Both describing the expected behaviours as well as holding people up to the bar - gently and firmly. I have not found a system to do this remotely or at scale.

Like everything in life competition is a fact of life. We compete for where we live, jobs, food, spouse, education, transport, etc. 
Sales is no different. We compete for clients, supplying our products and services, other salespeople withing and outside the company, etc. Often the reward system favours those who sell the most. Thus, by its nature selling will be competitive.
Like anything it is the degree to which one wants to take 'competition' and 'cooperation'. So first this has to be defined within the culture of the company. Not words but demonstrative action. Companies will have a different definition and its appication depending on what they sell and how they sell it. For example, If selling a high capital equipment $3 million item with a sell cycle of 12 months or more may require more cooperation than an over-the-counter item worth $10.
So first look at the culture of the company to determine the balance between 'competition' and 'cooperation' (Culture: The way we do things around here).

I've found success in gamification of KPIs. I read that someone had already covered developing realistic KPIs with the input of the sales team (excellent way to build collaboration, buy-in, and experience within the team) so I don't think I need to dive into that pool. Now that you have defined KPIs, build a competitive game around those KPIs. Remember that not all KPIs are individually driven. My suggestion is a points-based game related to the importance of each KPI. As the sales reps attain particular KPIs they amass points. Hang a reward on quarterly and yearly attainment. Be creative and remember that not all members of your team are motivated by money. Many years ago, when televisions were expensive, I worked with a mentor that gave TVs as a reward instead of cash. He put a plaque on each television containing the award title and the recipient's name. I was told that "People spend money tomorrow and forget about it. But the TV, the whole family will watch for years. Every time they sit in front of it, they'll remember the story of how their family member received recognition for an achievement." I'm not saying to use a TV as a reward, I'm just suggesting that the reward should have a lasting impact relevant to the achievement.