With my limited knowledge of the PF, and from what I'm reading above, I'd venture to say the Republic might have something akin to this in military structure:
Standing Navy and Army comprised of members from each Tribe. The statement above regarding Minmatar utilizing thieves and criminals to fill their ranks would also lead me to believe that these units could have easily formed Special Warfare units over their historical development. Taking a page out the Earth's past, many unsavory, if somewhat efficient units had dubious origins, and from these origins, specialized units were formed and set apart from the rank and file.
In regards to the individual tribes, no doubt traditional militias exist, but their combat efficiency and operational tempo may be largely dependent on the tribal leadership, and the actual role the militias fill in their respective societies may vary, whether they be truly combat capable, or merely ceremonial in appearance.
The existence of such militias would no doubt require a Unified Command that, while allowing each Tribe to operate autonomously, would no doubt exert command in times of Republic conflict. The Command would most likely be comprised of senior ranking tribal elders that carry either civilian or military titles.
The level of efficiency in the command's doctrine and structure can be debated, and perhaps exists as nothing more than paper knights chasing windmills, but the Amarrian Conflict in the least would have revealed to the Republic the need for a singular force capable of repelling any enemy and protecting the borders.
Which leads to an entirely different discussion on the nature of inter-tribal dynamics and political machinations. I think, however, that one ideal they agreed on, at least in principle, is the need for a combined force to serve the Republic. How efficient this is in practice, is anyone's guess.