Got it working with the Unity Runtime !
I had to uncheck the "Use indexes" option in Spine atlas settings. Damn.
I still have loads of errors, though...
Like this one :
NullReferenceException: Object reference not set to an instance of an object
SkeletonComponent.Clear () (at Assets/spine-unity/Assets/Spine/SkeletonComponent.cs:72)
SkeletonComponent.Update () (at Assets/spine-unity/Assets/Spine/SkeletonComponent.cs:113)
SkeletonComponent.OnEnable () (at Assets/spine-unity/Assets/Spine/SkeletonComponent.cs:294)
UnityEngine.GameObject:.ctor(String, Type[])
SpineEditor:CreateSkeletonAnimationGameObject() (at Assets/spine-unity/Assets/Spine/Editor/Menus.cs:73)
Thanks,
ben