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Heavy Infantry, Light Infantry (m/f), Ranger (m/f): available from the start

Court Mage (m/f), Mercenary (m/f): Gerard becomes emperor

Freelance Mage (m/f): develop the Incantations Lab

Imperial Guard (m/f): develop the Imperial Guard. Note that you'll need to annex a lot of territories to get this option (9 I believe), so don't fret if you don't see it.

Vagabond (m/f): Complete "Avalon's Cat Burglar." Make sure to choose to help Cat on the rooftops and use her secret route into the Canal Fortress and be sure to finish this in a single generation or you risk pissing Cat off and locking yourself out of the class. Note that you do not have to avoid encounters in the fortress after sneaking in.

Strategist: develop the Imperial University and speak with the Strategist then meet with the Nomads in The Steppe and agree to help them destroy the Landship. Return to the academy and ask the Strategist for help. Follow the questline from there and defeat Bokhon. Doing so will unlock Strategist and Nomad together. (There is a missable spell fusion tome in the Landship, Heat Hand)

Imperial Blacksmith (NEW TO THE REMAKE): develop the Smithy and complete "Pride of the Fabri Family" quest. There are five pieces of ore you need to find and they are in the Ludon Highlands, Salamat, Melu Desert, Mt. Comroon, and Termite Den. Return after finding them all and she'll join the retinue afterwards.

Martial Artist: Speak with the Nibel townsmen and then speak to the Martial Artist at Dragon Lair. Choose to help him regardless and make sure to listen to his request. Kill the first boss in the Fiend's Cloister and choose to return to the Martial Artist. Then go back to the dungeon and clear it to unlock the class. If you choose to ignore his request, you'll have to duel him and him showing up in your retinue from that point on is RNG, so just go with the guaranteed method unless you're experimenting.

Saigo Clansman: Complete "Retaking the Trade Route" and "The Mus' Migration" quests to unlock this class. Note that you may need to defeat Dantarg in order to complete the first quest but I've read that losing to him doesn't affect the quest.

Crusader (m/f): Complete the Cumberland scenario. there are multiple outcomes to this but the one I did involved not leaving the country until I finish the questline. Speak with the king, the siblings, choose to seek diplomacy with the siblings, speak with Sophia, then finish the rest of the quest to unlock them.

Nomad (m/f): Similar to above but they just require beating Bokhon to my knowledge.

Hunter: Reach Ostro from the Savannah (reachable from the Steppes past the Cumberland Gates) and complete "Nightmare in the Savannah". When the termites attack, choose to follow them into the hole and beat the Termite Queen. Then report back to the Hunter to unlock him.

Amazon: (I have no idea if this class can be recruited with a male emperor, so please details the steps in the comments if you can): Make sure the Emperor is female and travel to Salamat (reachable from the Savannah). Choose to help the Amazon (bottom right of the map) and follow her into the village. Be honest with her and she'll join your retinue afterwards.

Desert Guard: Reach Teretuva and complete "Hunting the Wandering Lake" quest. You can choose to either parlay with Noel or fight him. Note that if you killed Rocbouquet, he will fight you regardless in a much stronger form. Keep in mind that the Wandering Lake appears at random designated points across the desert.

Levante Guard: Reach Yuyan and complete "Sekishusai, the Royal Retainer" quest.

Ninja: Reach Yuyan and you'll recruit her along the way of "Sekishusai, the Royal Retainer".

Diver: Complete "The Storm and the Sunken Ship" quest. This is related to Subier's questline but you do not have to fight him to unlock Diver. I believe you do need to complete "Constructing the Leon Bridge" and "The Mystery of the Canal (buy the seashell from the NPC in Saiga to make this easier)" to unlock the quest, but I'm not 100% certain.

Dancer (NEW TO THE REMAKE): Reach Atlanticus and complete "The Dancer of Atlanticus" and "Searching for a Mermaid" quests. (the items for these quests will take you to Toba, Lake Aqua, The Savannah, and the Mole Den. You'll also unlock Nereid along the way) Then use the mermaid potion three times at the designated location to trigger a generation skip. Eventually the game will tell you of a new dancer in Atlanticus. Speak to her and ask her to join you.

Corsair: Complete "The Corsair's Dilemma" and "Garon's Rebellion" quests. Keep in mind that there are multiple outcomes to this. What I did was knock out the Corsair guard in the hideout, enter the room with the Boatswain, then chase after them through Cape Chalier. After that, enter Nueno and ask to form an alliance. Then a couple generations later you'll have to finish "Garon's Rebellion" and the Corsair will join you afterwards.
Also note that there is a missable fusion spell tome related to this. Make sure to speak to the boat captain and choose to help him with the nearby mines. The Messina Mines that unlocks afterwards holds the tome for Arc Lightning. You do not have to report to the captain after completing the dungeon.

Salamander: Grab the stone boats from the Amazon Village first. Reach Comroon Island, complete "Calming the Volcano", wait a generation or two, then complete "The Volcano's Eruption". Do not go to Tsukijima until the quest is complete or else the volcano will erupt and lock you out of the class. Note that recruiting Salamander will lock you out of recruiting Diviner and acquiring umbral magic.

Diviner: Grab the stone boats from the Amazon Village first. Reach Comroon Island, speak to the Diviner for "Tsukijima's Grimoire of Eld" quest. Then complete "Calming the Volcano", wait a generation or two, and accept but do not complete "The Volcano's Eruption." Let the volcano erupt by going to Tsukijima. You might need to speak to the Seer at the citadel first, then travel to the island that appears and complete "Secret of the Emerged Island". You'll also unlock umbral magic this way but you will be locked out of unlocking Salamander.

Mole: Similar requirements to the Hunter. Save the molesfolk from the termites (you'll be doing this along the way when unlocking Hunter), beat the Termite Queen and speak to the moles after. Then continue the story as normal until you reach a generation skip. Soon enough the game will tell you to check in on the moles and they'll join your retinue after that.

Nereid: You'll unlock this along the way of Dancer. Speak to the Nereid in Lake Aqua (reachable from the Ludon Highlands) for the water, then visit the Mole Den and speak with the crafter there. Travel to Avalon to choose to rest at the Inn to collect moonlight for the mole. Return to the Den and speak to the crafter again for the Moonlight Comb. Then go back to Lake Aqua and use the comb to unlock her class.

Iris: Reach Yuyan and complete "The Iris and the Minstrel" quest. Note that this quest will take you all over the world map to complete it. Recruiting the Iris also unlocks a possible quest completion route for Wagnas.

Autamaton (Coppelia): Complete "Constructing the Leon Bridge" and "The Mystery of the Canal." After that, speak to Hiraga at his home to recruit Coppelia. If the quest doesn't appear, you might have to wait a generation.