Shake the can of coconut milk or stir it well (this creates a consistent thickness, since the fat often solidifies at the top of the can).
In a 3- to 4-quart saucepan or wok over medium heat, simmer 1/2 cup of the coconut milk, stirring occasionally, until reduced by about half, 3 to 5 minutes. It will get very thick and shiny and may or may not separate; either is fine.
Add the curry paste, whisk well, and cook, continuing to whisk, for 1 minute. Whisk in the broth, sugar, fish sauce, and remaining coconut milk. Bring to a simmer over medium-high heat.
Add the carrots, onions, and ginger slices and continue to simmer, adjusting the heat as necessary. After 2 minutes, add the bell peppers and continue to simmer. After another minute, add the fish and simmer for 1 minute more. Add the bok choy and simmer until everything is tender and cooked through, about 1 minute more.
Remove the curry from the heat. Season to taste with more sugar and fish sauce, and add the cilantro. Transfer to a serving bowl (or serve right out of the pot). Remove the ginger slices or tell your guests to eat around them. Garnish with the cilantro, coconut milk, and lime wedges.
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