Are there any animal species other than humans & chimps that engage in tribal conflicts?
In the introduction of an article by Gavrilets and Fortunato, the authors give (with references) some examples of intra-specific group conflicts:


  Social organisms living in stable groups often engage in aggressive interactions with conspecifics from neighbouring groups over territory, mating opportunities and other resources... Examples include border patrols in chimpanzees, raiding parties in spider monkeys, clan wars in hyenas and between-group fights in lions, free-ranging dogs, meerkats, Capuchin monkeys, blue monkeys, black howler monkeys and ring-tail lemurs.


Whether these conflicts (or others, such as ant colony conflicts) meet your definition of "tribal conflicts" you will have to decide.