I have wanted to try running combat phases (one of my previous LL DMs ran it and it seemed interesting) but decided to stick w/side to group initiative instead. to me it keeps players engaged w/other players when planning their actions, freaks them out when enemies go first, and quickens combat a decent amount.
Its actually pretty quick.
It usually goes like this:
Before initiative: any spells being cast?
Each side rolls 1d6 initiative.
Side with initiative:
1. All characters check morale (NPCs only)
2. All characters move (advance or withdraw only, no fancy maneuvers)
3. All characters resolve missile fire
4. All characters resolve spell casting
5. All characters resolve melee attacks
Side without initiative repeats 1-5 above.
On a tied initiative, all characters (PC and NPC) go through 1-5 together
It tends to sound like this:
"Movement phase - GO! Okay, everybody where they want to be? Missile phase - GO!" and everyone rolls to attack and I apply damage. Then I say: "No spells. Melee phase - GO!" and everyone rolls to attack and I apply damage.
The only time it gets bogged down is with tied initiative. Then I have to proceed through each phase, character by character with the NPCs/monsters mixed in. I could just move monsters at the same time as player characters in one big group, but it can become chaotic, so I proceed down the list alphabetically or however characters are ordered in Foundry.