[AlyssaRowan]

There are usually procedures for that; varies a lot from place to place.

Here in the UK, you can go before a sort of judicial committee to do it, the Gender Recognition Panel; costs like £140ish. It's kind of a pain; you need to be at least 2 years in full-time, it does help if you've had SRS or at the very least are going to (and they're going to want medical reports), and you'll need two medical practitioners (one of which must be recognised in a list as a psychiatrist or psychologist specialising in gender dysphoria) to testify that yes, you are trans, yes, really you're not going to change your mind, and this is what you've done/are going to do about it. If you pass (and it's a little bizarre and intimidating that they're judging this), you get a new short birth certificate (and can apply for a long one, and thereafter a passport under your new official identity); you then have two proper birth certificates, the historical document you were born with, and a brand new one listing only your target gender (and name). The fact that you have done all this is confidential (unless you tell someone, obviously).

Your mileage will vary, naturally.

[ekianjo]

Thanks for the very complete answer. I had no idea there was such a procedure. Now I wonder how it works in Japan where I live, since such minorities are far from being commonly accepted here and they may have more hurdles to get to the same point.