Do native speakers of Japanese write romaji the same way native speakers of English would, or do they do it differently, possibly using the rules that they'd use for writing hiragana, katakana, and kanji?
This page shows an example of typical "proper stroke order of alphabets" which Japanese students may see in basic English textbooks:
See the stroke order of A, M and W, for instance. Do you think writing M with four strokes is odd? According to this question, Japanese Ministry of Education seems to have determined "the proper stroke order" that way long time ago. Some people seems to believe that this stroke order was influenced by the kanji writing system in which upward strokes occur rarely, but I don't know if that's true.
In reality, however, the stroke order is not considered important even at school, as long as Latin alphabets are concerned. Most people stop writing
W with four strokes very soon. This does not necessarily reflect how people usually write these alphabets in Japan.
I know very little about how alphabets are written by hand outside of Japan, so I'd like to see answers from other people. As a starter, this Wikipedia article summarizes regional handwriting variations of alphabets and numerals.
When I was a child in first grade, I learned how to write an "M" / "W" with four(4) strokes. A 6 year old child has problems with eye-hand coordination to get spacing correct on the "M" and "W".
Since traditionally the Japanese student did NOT learn how to write English characters until 4th grade, I would assume the Ministry of Education in Japan, just cookie cuttered some English learning program out of England, as a 9 year old does have more control over strokes.
Way back when as a French Canadian elementary school student, I don't remember stroke order being given a great deal of emphasis beyond serving as a convenient common ground when initially teaching how to write the letters, first for block letters in first grade, and later for cursive letters in fourth grade. After that, most people tended to develop their individual style that included a mix of block and cursive letters, written in whatever order was most comfortable for that person.
This contrasts strongly with the importance of correct stroke order in Japanese, where it matters not only for kanji, but also for hiragana and katakana. I suspect that this mentality was carried over to the roman alphabet when it became important to teach it in Japan, and its influence still continues: when Japanese kids start junior high school and English first becomes a full-time part of the curriculum, they are made to learn roman letters again, with the proper stroke order.
When I worked as a teacher at a cram school, it would trip me up at times because the official stroke order doesn't always match the way I got into the habit of writing the various letters when I was still in school.
The way Japanese people write the letters of the roman alphabet, then, will match that of a subset of native speakers of other languages using that alphabet, but I think you'd be hard pressed to find a common stroke order for every single letter among a large majority of those native speakers.