We do it by treating our daughters as a team. If one or the other behaves badly, then they both suffer consequences. They are 9 and 6, so old enough that they understand this (wouldn't recommend it for toddlers!). It works well, and they feel incentivised to behave kindly towards each other and work towards shared goals.
They don't feel that they get rewards or attention from us as adults by getting one over their sister. Of course, individual achievements are awarded, but we also encourage them to celebrate each other's success. We've introduced Sister Points that get granted or revoked when they are supportive/mean to each other.
When they get to certain milestones (50, 100, 200 points etc) they level up as sisters (they're currently Level 2 Sisters) and we go on a family day out (latest one will be an open top bus ride round the city).
I think we'll have to change tactics when they get to be teenagers, but so far it is working well. I had a really good, supportive relationship with my brother when I was young, and I want my two daughters to have the same.
Lindsay, you're a gem, this is exactly the advice I needed. We will begin "Sister Points" asap - they learn to make healthy choices and become empowered in the process.