First of all, the Mishnah:
מתני': ואלו יוצאות שלא בכתובה. העוברת על דת משה ויהודית. ואיזו היא דת משה? מאכילתו שאינו מעושר, ומשמשתו נדה, ולא קוצה לה חלה, ונודרת ואינה מקיימת.
ואיזוהי דת יהודית? יוצאה וראשה פרוע, וטווה בשוק, ומדברת עם כל אדם. אבא שאול אומר: אף המקללת יולדיו בפניו. רבי טרפון אומר: אף הקולנית. ואיזוהי קולנית? לכשהיא מדברת בתוך ביתה ושכיניה שומעין קולה.
And these are the women who leave [can be divorced] without a ketubah: One who transgresses against the law of Moses or [the law of] Yehudit/Jewish women. And what is the law of Moses? She feeds him [her husband] [food that] isn’t tithed, or who has sex with him while she is in Niddah, or who doesn’t take challah, or who makes a vow and doesn’t fulfill it. And what is the law of Yehudit? One who goes out with her head uncovered/wild, or who spins in the marketplace, or who speaks with all men. Aba Shaul says: even one who curses his parents in his presence. Rabbi Tarfon says: Even the Kolanit. And what is a Kolanit? When she speaks within her house, her neighbors hear her voice.
I have conveniently bolded the most relevant piece of this Mishnah, and left you the rest for context. One significant thing to consider is that where this version (following the print version) says דת יהודית, the manuscripts say דת יהודים, making it more like "the laws/ways of Jews"- the way that respectable people behave. That's why we're not looking for who Yehudit was, and why she was significant enough to be a parallel to Moses- she isn't a person, she's the way the society behaves.
So the Mishnah implies that cultural violations are enough to allow a divorce where the woman doesn't get her monetary settlement. And these violations include violations of the dress code and the behavioral code. For our interests, it leaves the requirement for a married woman to cover her head as a very strong cultural norm, which gets included in this religious law code. So from the Mishnah, it seems to have this ambiguous social/rabbinic force.
But stay tuned- the Gemara is about to (as usual) complicate matters.
Then, the relevant pieces of Gemara:
ואיזוהי דת יהודית? יוצאה וראשה פרוע:
ראשה פרוע דאורייתא היא, דכתיב (במדבר ה, יח) ופרע את ראש האשה. ותנא דבי רבי ישמעאל: אזהרה לבנות ישראל שלא יצאו בפרוע ראש. דאורייתא- קלתה שפיר דמי, דת יהודית- אפילו קלתה נמי אסור. אמר רבי אסי אמר ר' יוחנן: קלתה אין בה משום פרוע ראש הוי בה רבי זירא. היכא? אילימא בשוק- דת יהודית היא, ואלא בחצר אם כן, לא הנחת בת לאברהם אבינו שיושבת תחת בעלה. אמר אביי, ואיתימא רב כהנא- מחצר לחצר ודרך מבוי:
And what is the law of Yehudit? One who goes out with her head uncovered/undone. An uncovered/wild head is a Biblical prohibition! As it says (Numbers 5:18): and he uncovers/musses the woman’s head. And the house of Rabbi Yishmael teaches: [this pasuk is] a warning to the daughters of Israel, that they should not go out with their head uncovered/wild. Biblically, a basket is also permissible [as a head covering], according to the law of Yehudit, a basket is also forbidden. Rabbi Asi says in the name of Rabbi Yochanan: wearing a basket is not considered to be an uncovered head. Rabbi Zeira spoke about this statement [of Rabbi Yochanan]. Where [is he talking about]? If you say in the marketplace- this is a matter of the law of Yehudit! Rather, would you say in the courtyard?! If so, you haven’t left a single daughter of Abraham our father who can remain married! Abaye said, and some people say Rav Kahana said, [the situation is] from one courtyard to another, by way of an alley.
This Gemara takes on the Mishnah's assertion that head covering is only a rabbinic or social requirement, and points out that since it's mentioned in the Tanakh, it must be a biblical requirement. Having established that, it begins to establish two levels of requirement. The first suggestion is that the difference has to do with how much territory is covered: a biblical requirement that is satisfied with a very basic head covering- the level that an indented basket would cover- that is biblical, and a rabbinic requirement for a larger covering.
The other option establishes that location/public access is relevant (as to whether or not the basket is a sufficient covering). It points out that women aren't accustomed to covering extensively in the courtyard- meaning at home. So if extensive covering can't be needed at home- if so, no marriage would be left intact. A classic example of halakha following communal practice. The end is a compromise position, to harmonize, leaving alleys as the smallest place where full covering is necessary.
How does this sound to you? What are your reactions?