For full backward compatibility add it to constructor also:

class _StreamingResponse(Response):
    def __init__(
        self,
        content: AsyncContentStream,
        status_code: int = 200,
        headers: typing.Mapping[str, str] | None = None,
        media_type: str | None = None,
        background: BackgroundTask | None = None,                <<<
        info: typing.Mapping[str, typing.Any] | None = None,
    ) -> None:
        self.info = info
        self.body_iterator = content
        self.status_code = status_code
        self.media_type = media_type
        self.background = background                             <<<
        self.init_headers(headers)

This is because some middleware may be referring to that field. Or below will work also

self.background = None

No users should be calling the constructor unless they do something like type(response)(...) which is not something we need to support.

Agree, this aligns with the class being marked as private. Then please include

self.background = None

For full backward compatibility, any response object injected into BaseHTTPMiddleware based middleware needs to have the background field (default None)

Ok I see your point now with the edit. Good point. Added.

@Kludex this is another reason to move the background task logic out of responses and a good reason to rework our Response inheritance so that Response and StreamingResponse both inherit from a BaseResponse that has the API and initialization of common bits.