Does Typescript support mutually exclusive types?

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As proposed in this issue, you could use conditional types to write a XOR type:

type Without<T, U> = { [P in Exclude<keyof T, keyof U>]?: never };
type XOR<T, U> = (T | U) extends object ? (Without<T, U> & U) | (Without<U, T> & T) : T | U;

And now your example works:

interface Person {ethnicity: string;}
interface Pet {breed: string;}
function getOrigin(value: XOR<Person, Pet>) { /* ... */}

getOrigin({}); //Error
getOrigin({ethnicity: 'abc'}); //OK
getOrigin({breed: 'def'}); //OK
getOrigin({ethnicity: 'abc', breed: 'def'});//Error

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