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Tsukiji vs Toyosu
Both are real. They are not the same place, and they answer different questions.
The short answer
Go to Tsukiji to eat and walk. Go to Toyosu to watch the tuna auction from a viewing deck.
What actually moved
In October 2018 the wholesale market (the 'inner market') closed at Tsukiji and reopened in Toyosu, about 2.5 km away on Tokyo Bay. The tuna auction went with it.
The Outer Market did not move. It is a district of privately owned shops and restaurants next to where the old market stood. It is open today. Our atlas lists 325 of them from the official market association directory.
Tsukiji today
Narrow streets, shops that open before dawn and close after lunch, knives, dried bonito, tamagoyaki, sushi counters, seafood bowls. It is loud, close, and finished by mid-afternoon: 134 of the 325 listed shops close by 2 PM.
Toyosu today
A large, clean, purpose-built market on reclaimed land. Visitors watch the auction from a corridor, then eat at restaurants inside the complex. It is orderly and indoors. It is not a street market.
If you have one morning
Tsukiji. If you have two mornings, do the auction at Toyosu first (very early), then take the train or a taxi to Tsukiji for a late breakfast while the streets are still open.