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I am planning to accompany a friend on an overseas trip, where my friend will be undertaking several medical procedures to finish transitioning gender. These procedures include the actual gender reassignment surgery, and some plastic surgery.

His passport currently says his name is "John" and he is male. After the procedure, my friend's name will legally still be 'John', but her gender will be female, and her physical appear will also change. (Here I am using the word "gender" to refer to external genitalia, body shape, etc.)

Should we apply for an emergency travel document in order to return home, after the procedures are completed, given the discrepancy in appearance and gender with the official passport? Or will it still be possible to return on the original passport, post-operation, if my friend carries extensive documentation about the procedures performed?

(Both of us will be travelling on US passports.)

You do not need an emergency travel document.

Until you announce to the USA government that you have completed transition and provide documentation, they will not unilaterally change your sex in the records.

Thus your travel documents with previous sex still holds valid until you officially effect those changes when you return to the USA.

Your sex is not officially determined by your appearance. We are long past that period of stereotyping based purely on looks. There are many people who look like the opposite sex. Also there are many people who look radically different from their passport pictures including crossdressers who have not yet reassigned their gender.

In conclusion, regardless of how your friend looks now and how different his/her appearance is from what is on her official documents, he/she will be let in if he/she can prove with documentation that she/he is who she/he claims he/she is. Yes you will spend a little longer at immigration on arrival and when checking in at the departure airport however this is nothing new.

REFERENCES

https://transequality.org/know-your-rights/passports

In June 2010, the State Department announced a new policy to issue
passports that reflect a person’s current gender when either a
previous passport or other personal documentation presented by an
applicant reflects a different gender.

Under the new policy, a transgender person can obtain a passport
reflecting his or her current gender by submitting a certification
from a physician confirming that he or she has had appropriate
clinical treatment for gender transition. This policy replaces the
Department’s old policy, which required documentation of sex
reassignment surgery. In January 2011, the State Department made
further improvements to its new policy.

You will not need an emergency travel document.

First, the US does not have a single concept of "legal sex". Your friend's old passport will be accepted as long as it is valid, although they may have problems if their legal name does not match their passport.

Changing name and gender are two separate processes in the US. In order to change their name, your friend will have to request a name change through a court in their US residence. In order to change their sex listed on their passport, they will have to follow the process the state department has defined, which does not require surgery. This is separate of every other legal gender change process, including birth certificate, driver's license, and employment.

Finally, their gender, and what pronouns you should use for them, is not defined by their genitalia. You should use the pronouns your friend has asked you to use at all times, and not make up your own rules.

Source: personal experience. I traveled to the EU presenting female with the wrong gender and old name on my passport, and have since had experience going to several countries with my updated passport in my new name and with female on my passport.