Show of Hands

It's a pleasant thought, but I think human nature would just have us regrouping them all too soon, whether we called them the same things or not.

Agree. In fact, for most elections, we have that with our top two primary system here in my state. Sometimes the general election is between two candidates in the same party:
wei.sos.wa.gov/agency/osos/en/voters/Pages/top_2_primary.aspx

From what we saw of historical republics, even without official parties, people will organize themselves as such.

Judges do that and it's just a shell game.

It would be nice. However even de facto, we would still consider them reps or dems based off their choices and the ideologies would flock together.