There are those who say that one must make the ציצית [strings] from the same color as the טלית because it is a הידור מצוה where the בגד and the strings are the same color, and so are the מדקדקים נוהג [1]. This is based on the רמבם who writes [2]:
טלית שהיא כולה אדומה או ירוקה או משאר צבעונים - עושה חוטי לבן שלה כעין צבעה, אם ירוקה ירוקין, ואם אדומה אדומין.
A טלית that is entirely red or green or other colors - we make its white strings similar to its own color: if green - green; and if red - red.
The רמבם, in turn, is based on a ברייתא [3] which teaches that a טלית colored entirely תכלת [which is blue] can hold strings of any color (besides for indigo [which is blue]). This is because if a טלית is made of תכלת and its strings are blue - then someone may think that all the strings are blue from תכלת (which is derived from a sea creature and would be כשר) rather than blue from indigo (which is derived from a plant and would be פסול). The ברייתא is teaching us that ציצית strings do not have to be white per se, but rather a different color than תכלת.
The רמבם there continues:
היתה כולה תכלת - עושה לבן שלה משאר צבעונים חוץ מן השחור מפני שהוא נראה כתכלת וכורך חוט אחד של תכלת על הכל כדרך שעושה בשאר ציציות שאינן צבועין.
If a טלית was colored entirely blue - then one makes its strings (which would otherwise be white) from all other colors besides for black [4] which is similar to תכלת; one then wraps one string of תכלת around the other strings the way one does for other ציציות that are not colored.
There are those who disagree on this point, however, saying that even for a טלית made entirely of blue fabric one can nonetheless wear blue strings. To that end, all of כלל ישראל is נוהג to wear a white טלית with white strings, thereby avoiding any מחלוקת הפוסקים [5]. An additional reason to wear a white טלית is because Hashem's clothing, as it were, is described by דניאל [6] as being white like snow.
אשכנזים, specifically, who were not always מקפיד to wear a white טלית were still נוהג to make the ציצית strings white regardless of the color, because אשכנזים are only מקפיד that the material should be the same but not the color [7]. One should not change this מנהג [8]. Even, however, someone whose טלית גדול is of whatever color (not specifically white) - nevertheless, a מדקדק should make sure that at least his טלית קטן should be white, so as to satisfy all opinions where he makes the strings white [9].
[1] לבוש שם סעיף ה ,טז שם ס"ק ז ,מחבר או"ח ט:ה
[2] ציצית ב:ח
[3] מנחות מא:
[4] The רמבם here is לשיטתו in that the רמבם (ציצית ב:ב) relates that the blood of a חלזון [the sea creature from which תכלת is derived] has blood that is black like ink. It is unclear where the רמבם obtained this information. [Some say the רמבם found a חלזון from which he made תכלת, and that this חלזון's blood was black. However, in the opinion of this writer, this is not מוכרח from the context in which the רמבם lists the criteria for determining what a חלזון is.] The classic נושאי כלים are silent on this point. To that end, I turned to ר חיים Kanievsky's קרית מלך, which offers sources for the רמבם in all the places where the classic נושאי כלים did not do so themselves for whatever reason.
Here, ר חיים references פתיל תכלת which brings 'somewhat of a ראיה' from the ירושלמי (נדה ב:ז) where sea creatures are described as having blood which is black like ink; and where the ברייתא (מנחות מד.) described a חלזון - it does so by saying 'ברייתו דומה לדג' [the structure of its body is like that of a fish]. The רמבם, according to ר חיים's interpretation, seems to be applying the principle of a חלזון's body structure being like that of a fish to the blood of the חלזון as well.
[5] בח או"ח שם בשם הרי, וכתב הראבד על הרמבם שם שקושיתו אינה קשא עיי"ש.
[6] ז:ט
[7] ע"פ תשובת הרשבא ג:רפ לבוש או"ח שם; תרומת הדשן סימן מו
[8] רמא או"ח שם
[9] טז שם ס"ק ח
