The Codex meta is still very new. However I think it's developed enough that there are a few "known strong" multicolor combinations (beyond the fairly well explored in-color synergies). I've made a list of some of the best-known ones below.
All of these have found a measure of sustained success for multiple players in tournaments and pbf games. More are still being found - if I'd written this a month ago, the first entry would have just been [Past]/Peace, but EricF's CAMS run with [Future]/Peace has shown that to be just as strong a synergy.
The Peace Engine plan of Flagstone Garrison + Drill Sergeant + Overeager Cadet and other cheap units is already quite strong. Purple starter adds in several elements that make it even stronger. Battle Suits beefs up the Cadets so that they can trade favorably at Tech I, Nullcraft is a potent target for Drill Sergeant's +1/+1 runes, and Mox keeps the deck thin for the draw engine to go through more reliably. Mox is also one of the best targets in the game for Boot Camp - an indestructible 3/3 or 4/4 is quite oppressive in a lot of matchups.
Past is the most common variant (especially as [Past]/Peace/Anarchy). Stewardess of the Undone helps you keep tempo while getting to Tech II, and Undo and Rewind can counter a wide range of strategies. Playing Future instead brings Knight of the Conclave as another free unit to keep the engine going, Vir's abilities to make sure your draws line up how you need them, Assimilate to counter other building/upgrade based strategies, and Unphase to sneak a runed up Drill Sergeant through the patrol zone. Present has the least synergy, since it relies on its own powerful (and expensive) Tech II units.
Crashbarrows and Shoddy Gliders are cheap and powerful, but because they don't stick around on the board they are unreliable and leave you with no patrollers. Garth's midband draws and maxband ability help you play one when you need it and Graveyard helps you play it again. Skeletons and Lich's Bargain can stuff the patrol zone to blunt your opponent's counterattack. If they survive, Drakk's midband ability can give them a potent counterattack of their own.
Another Blood Tech II synergy (and often combined with the previous one as [Necro]/Blood/Truth). This time instead of replaying the same unit, you get multiple copies of it using Quince's mirrors. Hitting with 3x Crashbarrows in the midgame is usually a board wipe and a tech building kill, which is often game ending. Future/Truth is a similar combination, swapping in Omegacrons for Crashbarrows.
Vandy + black starter is already a strong combination, but every one of River's regular spells makes her more potent. At Tech 0, Harmony synergizes excellently with Dark Pact, which Vandy already wants to play a lot. With Vandy + black starter's already strong early pressure , a quick Heroes' Hall and Harmony + Dark Pact. can end the game before it really starts. At Tech I, Two Step turns a Gargoyle into an oppressive tech destroying machine as long as River stays alive. At Tech II, Discord is the most reliable way in the game to unshackle Terras Q.
This combination is built around Might of Leaf and Claw. MoLaC's drawback is that it requires board presence to activate, but spending 3 gold and a card on it at Tech II makes it hard to do that. Anarchy gets around that by bringing in the most access to haste out of any spec. For 7 gold, Zane + Surprise Attack gives you 8 damage and 3 runes on MoLaC from an empty board. Add in all the haste options from the red starter (Mad Man, Makeshift Rambaster, Charge), and this combo has an excellent chance to activate MoLaC. That is usually game ending unless the opponent has a hard counter to upgrades, which only a few specs do.
There is a whole thread about the synergies in this combination, which cover a wide range of strategies. For a Growth Tech II plan, Bird's Nest is a strong complement. Blooming Ancient gives runes every time the birds are resummoned, and the birds themselves are a strong carrier of those runes. The birds and Rook are also fairly reliable sources of attacks for MoLaC, since they are hard to get rid of once they hit the board. For a Strength Tech II plan, Doubling Barbarbarian loves loves loves Arg's spells - a single Dinosize let's him win a trade with every non-flying Tech III in the game and still have damage left to overpower to something else. At Tech I, Galina Glimmer has a strong synergy with Mythmaking, which makes her a 4/4 for 1. Since none of these synergies depend that strongly on either the White or Green starter, this combo can be easily combined with the previous one as [Anarchy]/Strength/Growth.