Who is a Jew

Judaism has no overall governing authority. So how do you know who is a Jew?
There are all kinds of groups claiming to be Jews who make contradictory assertions:

1. Talmud and rabbis are voices of authority vs Talmud and rabbis have no special authority
2. “If your mother is a Jew, so are you” vs “if your father is a Jew, so are you.”
3. Cut necessary vs no cut necessary
4. It is primarily a cultural thing vs primarily a religious thing vs primarily an ethical thing
5. Must support Israel vs support for Israel not important

It clearly has no particular biological connection – there is no biological test which demonstrates conclusively that someone is Jewish, so claiming Judaism is a “tribal” thing doesn’t make much sense (besides, that’s really just an ethnographic cultural claim, so back to (4) above).

If no one has the authority to even define Judaism and who is or who isn’t part of it, does Judaism even exist? Or is it just a pleasant fiction, like pretending to be Wiccan?