There are a total of 10 factions within the area the MC is in. Here is the list of factions with a small history behind them.
Sorry if it's a bit scoffed as I'll try to do my best to explain each faction in the simplest manner.

1st Faction - Brotherhood of Steel - Manila Chapter

Formed in 2095 under the leadership of Elder Zacka, they hold one of the most secured l, but contested territories in all of Luzon proper numbering around 600 men/women strong in their army.

Their numbers back then were small and had survived the Chinese missile attacks and washed ashore near the Palawan islands.

At first, they tried to contact fleet command, but when no one responded, they tried to contact the American government or any government and despaired when they realized no one had answered their calls.

For 18 years they tried to survive the apocalypse and eventually, after 18 years, they founded what's known as the Brotherhood of Steel when they noticed that the American people along with some of its military were still alive. 

Well barely alive and copied their name with the very first brotherhood of steels that popped up within America.

Reinvigorated, they set off to the capital of Old Manila where it was nuked twice and established a foothold there they started to help the people who were surviving and killing off any monsters and bandits alike and founded New Manila under the Banner of the Brotherhood - Manila Chapter.

Today it now stands as one of the oldest factions in the Philippines and one of the most respected but hated enemies of the Chinese communist forces and others.

They controlled not only New Manila, but Navotas, and Pasay City as well and were planning to expand toward Makati when they encountered Chinese forces and to this day, they are still at a stalemate.

The stalemate lead to negotiations and eventually the BoS retreated for now to consolidate their forces and fortify their borders against the communist threat for now.

But the new elder is keen to get his revenge of their fallen brothers and sisters to the barbaric ways of the Chinese. And they will have blood.

2nd Faction - Brotherhood of Steel Deserters - Quezon Chapter

Founded on 2270 under the leadership of High Paladin Gerald. Gerald a veteran of the war between Manila and Makati where his forces were nearly slaughtered when they occupied and was ordered to defend Makati City Hall from the Chinese forces.

And after 4 hours of hell, reinforcements finally arrived, but it was too late as Gerald and only 12 Paladins and 20 Initiates were alive out of 100. A devastating loss considering how long Gerald and his men held the city hall.

It didn't help that a month after, his men would only die in vain when Elder Hollen signed a humiliating treaty against the Chinese. Stating that, they need to build up their numbers for now and fortify their borders.

But Paladin Gerald wasn't having it, he tried numerous times to convince the Elder along with the other high paladins that were present in the hall, but his words never got through and in 2270, he took a decent amount of paladins, scribes, Initiates and headed to Quezon City to start anew and fresh.

Elder Hollen didn't take Gerald's betrayal well and branded him and his men under him a traitor to the chapter and declared them as deserters.

Now in 2277, High Paladin, now Elder Gerald established himself as the leader of a small but decent-sized faction or army within Quezon City where Elder Gerald is heavily recruiting men, women, and even children for the upcoming war he has planned to take Makati.

3rd Faction - Blood Of the Damned.

No one knows how the Super Mutants came to be, or Giants as what the locals call them, they mostly reside in the most irradiated areas and underground sewers, tunnels, and even metros.

The more smart Mutants made a name of themselves and establish small factions and isolate themselves from human society.

The most civilized and sociable faction is the Blood of the Damned.

Numbering at 120 Mutants with it's leader name Rockface, they established a hierarchy that's similar to the olden days and made a name of themselves as a neutral faction but part time mercenaries.

This said faction is in Taguig where most of the small missiles that contained chemicals rained down on and most say that Taguig is heavily irradiated and can glow in different colors depending on where you are.

4th Faction - Chinese United Steel - Makati Chapter

Founded when the battle between the brotherhood and the Chinese forces are at its height, initiates, along with scribes and a paladin were captured and were slowly brainwash to work for their cause along with adding their own forces such as the stealth units.

Under the leadership of General Fuzue, this chapter seeks conquering all of the islands, then head to mainland china to reunite with their motherland and invade america once again.

5th Faction - Enclave (Asian Foothold)

Founded in 2200, this faction consists of old ghoullified American naval officers who dreams of bringing back the American dream by any means necessary and will even risk using nuclear weapons if need be to achieve that dream.

Stationed not so far from Subic Bay, a Carrier along with 3 Destroyers under the enclave are anchored not too far from the Subic Shore, watching the ongoing conflicts within the mainland and waiting for an opportunity to strike by using their missiles.

The key issue that they're facing right now is armament production along with their ships being so old now that it needs parts to fully sail again.

So for now, they're sending their spies and agents to keep surveying and watching the mainland for any movements that might disrupt their plans in the future.

6th Faction - Phillippine Liberation Republic

Founded on 2275, a recently early faction that has decided to go public after establishing their fortress or new government in Batangas.

They seek to restore the Philippines through military force and through democracy in their own way and vision.

They currently are trying to expand towards the east and south to gain more manpower to liberate Manila proper from the hands of enemies.

7th Faction - Nightingale Association

Founded in 2276, The Nightingale Association is a faction that consists mostly of people that are more neutral but want to help the people of the wasteland to have a better life via giving them ways to combat drug addiction, PTSD, etc etc.

This faction is lead by a woman named Nightingale who fancied herself to be the legendary Florence nightingale who conquered France for a time (She believes it) and in her vision, she want to eventually establish herself and her people that their faction isn't to be trifled with.

They are mostly moving from place to place as they don't really have a permanent home but they mostly move from north Luzon to south Luzon depending on the Nightingales decision.

8th Faction - Army of the Great Chang

Mostly founded by surviving Chinese civilians and Chinese forces that were in the Philippines at the time, the faction upholds and follows their leader Shang religiously and dims everyone a heretic and must be cleansed by fire in order for heaven to judge them worthy and be reborn as a Chinese.

This Army cult has mostly established themselves in Tagaytay and remained there to this day, brainwashing the locals, and amassing their zealots to someday march north to help their brothers in Makati to finally end the brotherhood Manila chapter for good.

9th Faction - Europa Eternal

Founded by European locals and foreigners after the war, this faction is but a small faction, but focuses around on business and controlling the money within the Phillippine wasteland.

The flag is their symbol of peace through power. meaning if you make them their enemy, they will use not just their assassin's to come after you, but to make you as an individual or owner of anything to make you suffer economically.

They mostly control the money that's flowing by half on Manila proper and many other small communities throughout  Luzon proper.

10th Faction - Luzon Freedom Fighters.

No one really knows how they're founded as they rarely show up in anyone's radar and they lived in the thick forest of Sierra Madre where eldritch horrors await to the people that dare venture inside the forest.

Most faction begrudgingly respect this faction since they have the balls to live in such a dangerous forest and lived to tell the tale.

They mostly fight for the freedom of people and do constant raids on slavers and most factions they come across.

They don't care on who's their friend or not, they only care about their portion of the goods that they're aquired along with the slaves they've freed.

There's rumors though that the leader of the freedom fighters is being lead by an AI, but who would believe in such a lie?

Fortuna Industries - Partner of Vault TEC.

Fortuna Industries partnered with Vault TEC in 2050 and ever since then, Fortuna Industries has dominated the Asian market as both American and Asian tech were sold like hotcakes before the resource wars started.

In 2060 to 2070, Vault tech negotiated to Fortuna to establish Vaults similar to the American vaults that they're making and Fortuna Industries agreed.

Multiple vaults within the year 2060 upto 2077 numerous vaults were completed with the most vaults being completed in the Philippines with 65 Vaults.

In total, Fortuna Industries nearly bankrupting themselves by building around 300 Vaults with 65 being in the Philippines as no one would target the Philippines since it's not an important strategic target.

In October of 2077, Fortuna Industries were busy adding the final touches of the vaults when the news reached Fortuna Industries from Vault TEC that the American government along with the rest of the government all around the world will launch nukes in 2 hours 

In a panic, Fortuna Industries initiated the Gate Protocol where candidates that were chosen to enter the vault rushed inside to get a spot, some even tried to force their way in as people were terrified that this was the end of the world.

And it was, on that day, almost all of the nuclear missiles went airborne, heading towards their target.

For the vaults within the Phillippine islands, the vault dwellers as many come to name them would experience hell, and more hell when they finally see the surface after 200 years or more give or take depending on when they open the vaults.