I'm playing Lost Mine of Phandelver (LMoP) with my friends. When I read the adventure, I noticed that there are multiple opportunities to join different factions.
What happens if a character has an alignment compatible to multiple factions? Can he join all of them?
I'm aware that you can gain renown only for one faction at time, and if you decide to gain renown on another faction, the one you switch from will go to 0 renown, but it doesn't look like it's harmful to just join several and then decide where to rank (or at least I haven't found anything in the rules that would prohibit this).
Consider the DMG at p.22, "Renown":
That said, in Adventurers League it is only possible to belong to one faction at a time, and gaining renown in a second faction zeroes out renown earned in the first. See the Adventurers League Player's Guide at p.3.