The "reading" for your new Japanese name

Much like how an English name can be pronounced different ways for identical spellings depending on the person, Japanese can have absolutely completely different ways to pronounce a particular person's proper names. For example, the not uncommon woman's name 智子 can be read as:
- ともこ "Tomoko"
- ちえこ "Chieko"
- さとし "Satoshi"
- さとこ "Satoko"
- さとみ "Satomi"
- さとね "Satone"
- ともし "Tomoshi"
- ともみ "Tomomi"
- ともね "Tomone"
- ちこ "Chiko"
- ちね "Chine"
- ちえね "Chiene"
How do you know the way to read it? Most people will take am educated guess at the most common reading, "Tomoko", and be right for most of the cases. Japanese dictionaries will, given a certain sinogram proper name compound, try to identify as many known readings as possible, indicating how common they are and what readings are "for names only." But the only way to know for sure is to ask the person or look at a document that the person made with both their name & "reading" of the name. People with unusual or "difficult" readings for their name will often (but not always) have the reading in ルビ (small ruby font) above the 漢字 name, in 仮名 (Japanese syllabet), on their 名刺 (business/calling card).
A month ago I was at a wedding and I was surprised when I noticed the bride had the same 漢字 given name as my wife. She corrected me, saying her name was 『まなみ』 , not 『まみ』 , and yes, she gets that all the time as her reading is very unusual.
This is why almost all forms in Japan which ask for one's name will have an additional field above it asking for the reading: there is simply no way to guarantee a name is read a certain way, and customer service wants to read the name correctly the first time.
On the Application form for Naturalization Permission (帰化許可申請書 ), just like a 出生届 (Japanese birth registration form), you will specify both your new Japanese name — in any particular form of 漢字 or 仮名 (Japanese sinograms or syllabet) — and how to read it, in カタカナ (Japanese syllabet). However, you should be aware that the reading for your name will not be on your family register (戸籍 ).
The 読み方/ふりがな (sinogram reading) you put on the application will indeed go on your 帰化者の身分証明書 (Proof of identify of naturalized person), and this document will then be used to create both your 住民票 (local resident registery) and your 戸籍 (family register).
But only your 住民票 (local resident registry) will have the reading of your Japanese name on it. When your local resident registry changes due to a move, your sinogram reading does not get automatically transferred.
In the past, some Japanese had the reading for their name put on their 戸籍 by having a 傍訓 (margin note) added to their paper family register. The problem with this though is the 傍訓 (margin note) does not automatically propagate from register to register. If a derivative register is made due to a family event, for example, the margin note is not transferred, but rather stays on the old document and a new margin note must be added.
Starting near the end of 1994, though, this became no longer possible. Why? They began digitizing the family registers, changing them from a Japanese B4 piece of paper to a database entry printed on special A4 form paper. They are expected to finish the complete digitization of all family registers by 2015, but all new registers created after 1994 are now done digitally; adding a 傍訓 (margin note) is not possible on the digital versions.
Readings for naturalized citizens are important because they sometimes choose non-traditional names. If they are Chinese or Korean, they may use the same 漢字 that they used as a Chinese or Korean — though the character may be changed to 新字体 ("new" Japanese-style simplified forms) and/or modified so it corresponds to a "close enough" character in the Japanese family register character database. When they do this, they can choose any reading they want: they may choose a "Japanese style" way of reading or they can approximate their original Chinese/Korean name in 仮名 (Japanese syllabet). If they do this, most Japanese will not be able to correctly guess how to read their names, unless they happen to have studied how to read foreign languages like Chinese or Korean 漢字 .
Other foreigners who are not Chinese or Korean — such as Americans and Europeans — may choose to approximate or derive from their legal name in 仮名 (Japanese syllabet sounds) an 当て字 漢字 name which have characters that are correspond either in meaning, pronunciation, or spelling to their original name.
Transliterating foreign proper names into 漢字 , while common in China, hasn't been common practice in Japan for over a century. While some very common names have known 当て字 for them, most people arbitrarily mix the オン (old/ancient Chinese origin/derived) and くん (Japan origin) readings for characters with good meanings to make the characters "fit" — as close as possible. Some people with Chinese language knowledge pick the common 当て字 for their name used in China (which still does the practice because they don't have a common-use syllabet or use Latin letter names in the middle of Chinese text) and then arbitrarily assign the 仮名 (Japanese syllabet) 読み方 (reading) to that name.
The disadvantage to doing this, just like having an unusual name in the U.S. or any other country, is that most people will not be able to correctly read your name aloud without written or verbal guidance.
If for some reason someone needs proof that your name is "read" a certain way, you will need to provide "proof" that is not a 戸籍 (family register). Japanese driver's licenses, by the way, do not list the reading of one's name either.
Fortunately, your 漢字 name is a good enough match for identification purposes; organizations don't normally need the 読み方 for an identity match. The reading is used to sort names in a list (for example, a cell phone address book), though, because 漢字 dictionary order is too difficult — even though people who have finished compulsory education know how to do this.