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Aliens, robots, Biblical creatures disguised as a cute girl transfers into [protagonist] school is a cliche.

But, how do I really do it?

Assume,

Local skin tone, Accent, Mid-class and other average student physical characteristics. The being has sufficient common sense and mental health to not look out of the ordinary.

But, has no, ID card, birth certificate , basically in the eyes of the government doesn't exist. Like just pop out of nowhere. Like a bastard born child kept quiet or something.

How to solve this assuming mind-control and all manner of magic is out of the question. Actually does not have to be a super-natural in the first place.

What age is she?

Where I live undocumented children will have no problem getting into elementary school as it is considered a human right. She will have to be able to convince the school that she is a "proper" undocumented alien though, as opposed to an abductee or something. Any sign of abuse or neglect on her will get the authorities involved very quickly. Being fluent in a foreign language might help her keep up the appearance.

Higher education is a problem. She is gonna need a social security number and some means of identification. These can be forged however, and as long as she avoids scrutiny she will be fine, nobody will investigate. As long as all bills are payed and she plays "average student" well there will be little problem.

As another answer already mentioned though, she will have to have some people to act like her parents/guardians from time to time, for as long as she is supposed to be a minor. They do not need documents, but will have to act convincingly.

If the school is big enough to have a complex bureaucratic systems the only thing required is to someone from inside (or even a hacker) to add the new students to the system.

The teachers may not care about the student life outside school, so nobody will audit or double check the archives to see from where that person came from or ask if they were really suppose to be there.

That works even better in a big city or in a community without strong ties where parents of students don't get into each others business.

Identity theft

1) Find a cute girl of the required age who is about to transfer into said school.

2) Occupy her body or kill her and wear her skin.

3) Brainwash the parents if necessary. Chances are the changes in her personality can just be explained by puberty.

The question really becomes what forms of verification does the school require. Which essentially boils down to how strict does the person that "admits" students want to be. In some locations proof of being an adult that lives in district is enough to get ones "child" enrolled. So if the student had another entity, essentially a person of parental/guardian age that could show proof of residency it would be little issue to get someone in to school. As the district becomes larger or more sought after as a place to attend shool this likely become more difficult.

Getting someone into a school with a legal age helper would be much easier than a school age person being admitted on their own though. Imagine a 16 year old walking up to a school secretary asking to get enrolled. They are far more likely to be referred to a foster organization then be allowed to attend classes without some adult to vouch for them.

Greetings prinipal of prestigious private school. I am an eccentric billionaire and I'd like to enroll my daughter. She doesn't have any documentation because the documentation butler was actually an actor hired by my nemesis and I had to fire him. Hopefully you can arrange all of that for me. How many sacks with a dollar bill sign on them should I leave with you?

Have an ally with valid ID adopt the supposed child from a country with bad records, to then get valid ID for the country with the school in. Celebrities like Madonna do so frequently!

A couple of ways that do not involve hurting/killing anyone

1) Simply bloody yourself up and be found in an ally by the police. Pretend you do not remember who you are and the government will likely make a perfectly legal entry in the system for you.

2) "Immigrate" from a country with bad/no records, to a friendly country (like Canada). Again, the government will make a perfectly legal identity for you.

3) Grease the palms of the morgue staff to "forget" to file a death certificate for a Jane Doe (even better if her name is common). Assume this person's identity. As a magical/alien creature, you can likely trade fairy dust or ray guns easily.

4) Simply show up and act like you belong there. Several con artist have spent years playing the bureaucracy