Gemara

Question: According to the opinion that Adam had two Partzufim (faces), which faced forward?

Answer (Rav Nachman bar Yitzchak): Presumably, the male faced forward;

(Beraisa): One may not walk behind a woman on the road, even one's wife.

If he encounters a woman in front of him on a bridge, he goes to her side until he passes her.

Anyone who goes behind a woman in a river has no share in the world to come. (When she lifts up her clothing to keep it dry, he will see.)

Kidushin 81a: Rav and Rav Yehudah were on the road. A woman was in front of them. Rav urged that they go in front of her.

Rav Yehudah: You permit two proper men to be secluded with a woman!

Rav: Perhaps only people like R. Chanina bar Papi are considered proper!

Rishonim

Rambam (Hilchos Isurei Bi'ah 21:22): If one encounters a woman in the market, he may not walk in back of her. Rather, he runs and goes to the side or in front of her. Anyone who walks in back of a woman is a lowly ignoramus. One may not pass within four Amos within the doorway of a Zonah, for it says "Al Tikrav El Pesach Beisah."

Rashi (18b DH v'Afilu): It is disgraceful to walk in back of one's wife.

Maharsha (Berachos 61a DH v'Afilu): Surely, there is no Isur to walk in back of one's wife. It is only due to disgrace. It is improper for the more important one to walk in back. However, the Rambam forbids "excess talk with a woman" even with one's wife, lest he accustom himself to excessive relations. We can say similarly about walking in back of her. Perhaps Rashi explained that there is no Isur here, for each Partzuf faced a different direction. Even if the female went first, the male would not look at her. It is only due to disgrace. Rashi explained that one must pass in front of or to the side of a married woman, but not to one's own wife. Since it is only due to disgrace, we would not burden a man to go to the side. Rather, it discusses another's wife, and it is forbidden to walk in back of her.

Poskim

Shulchan Aruch (EH 21:1): If he encounters a woman in front of him in the market, he may not walk in back of her. Rather, he runs and goes to the side or in front of her. One may not pass within four Amos within the doorway of a Zonah.

Radvaz (2:770): In these lands, women are totally covered. There is nothing a man can look at. Even so, one may not walk in back of a woman. Chachamim did not distinguish. All women in all places are forbidden. Also, the Isur is not due to looking at exposed places. Chachamim taught that looking at the heel is like looking at the Ervah! Rather, it is lest he come to Hirhur (thoughts) through her walking and movement, even if she is fully covered. Surely, in the days of the Gemara women walked covered, and even so it was forbidden. One may not walk in back of even one's own wife, even though there is no Isur to see something uncovered on her. Rather, it is due to Hirhur. This applies even to one's wife. This is why we say that Mano'ach was an Am ha'Aretz. It is not enough to walk four Amos in back of her, for he is in line with her.

Question: Why is this worse than a Zonah's door? It suffices to distance four Amos from it!

Answer (Radvaz): One must distance four Amos from a Zonah in every direction. One must distance from other women only if he is directly in back of them. One may come within four Amos of them on occasions. Walking in back is forbidden even outside four Amos, for surely it will lead to Hirhur. One may walk in back only if he is so distant that he will not notice her walking and the motion of her limbs, it is forbidden. He must rush to pass in front of her, or go to the side so he will not see her movement. If not, he must wait until she goes in front [so far that he will not see her movement]. If there is another path, he takes it. If none of these are possible, and he is going to a Mitzvah, and if he delays he will lose the Mitzvah, he walks without lifting his eyes from the ground.

Be'er Heitev (2): One may not walk within four Amos in back of her (Terumas ha'Deshen, Beis Hillel). Keneses ha'Gedolah brings that the Radvaz always forbids if he is close enough to notice her motions, even where women are fully covered.

Pischei Teshuvah (1): The Radvaz connotes that the Heter to walk in back without lifting his eyes from the ground is only if he is going to a Mitzvah.

Beis Meir: The Gemara mentioned only on the road. The Rambam mentioned in the market. Seclusion does not apply there. It must be due to Hirhur. How does he explain Rav Yehudah? Rav Yehudah asked Rav why they must pass in front of the woman, since seclusion with two men is permitted! Perhaps he asked why they must pass in front. It should suffice to go to the side to avoid Hirhur. The Gemara did not mention going to the side on the road, for sometimes this is forbidden due to seclusion. It said to go to the side on a bridge, for many pass over it, and seclusion is not a problem.

Otzar ha'Poskim (16:2): Nesivos ha'Shabbos permits four Amos in back, just like one may pray more than four Amos in back of his Rebbi (OC 90:24). Rav Yehudah asked Rav why they must pass in front, since they were more than four Amos in back, so there is no concern for Hirhur! Kerem Chemed asked that if so, the Shulchan Aruch should have taught that four Amos in back is permitted! Rather, it is forbidden if one can see her, even outside of four Amos.

Einayim l'Mishpat (Berachos 61a:1): It is disgraceful due to suspicion, lest people not know that she is his wife. We say that a Chacham should not speak with a woman in the market, even to his wife, [daughter or sister], for not all know who are his relatives (Berachos 43b). This is only for a Chacham. Avos d'R. Noson (2:2) says not to walk in back of a woman in the market, even with his wife, due to what people will say. One must distance from what is ugly or close to this. Binyan Yehoshua (there) says that the concern is lest people not know that it is his wife.

Note: Hagahos ha'Gra there (9) changes the text to say that one should not walk with a woman in the market [even his wife].

Otzar ha'Poskim (16:4): The Shlah (Sha'ar ha'Osiyos Kuf) forbids even a blind man who realizes that he is in back of a woman. If Adam had two faces, surely the male walked forward, even though he could not see the female in any case. Avos ha'Rosh questioned this and rejected the proof. (Note: Rashi and the Maharsha hold that it is disgraceful, but not Asur, for the female to go first.) Sedei Chemed agreed, but said to be stringent anyway, for Sha'ar Efrayim (78) and Mahari Atyei brought the Shlah without comment.

Otzar ha'Poskim (16:5): Binyan Shlomo (6) says that perhaps even several men may not walk in back of a woman. Nesivos ha'Shabbos and Beis Meir clearly hold like this.

Tzitz Eliezer (9:50): Leket Yosher (376, b'Sof) brings from the Terumas ha'Deshen that nowadays one may walk in back of a Chacham's wife or one's mother, for we are not commanded so much about walking in back. This is astounding. Why is a Chacham's wife different? Perhaps he discusses a case in which he shows honor to her [e.g. to let her enter first. Nowadays this applies to buses...] He taught this together with going in back of one's mother. The Mitzvah to honor the Torah or one's mother overrides. However, regarding his mother, perhaps there is no concern for Hirhur. However, why does he conclude 'we are not commanded so much nowadays'? Perhaps he held that there is reason to be lenient when she is totally covered, like the Radvaz' Hava Amina, and he relies on this for honor of Torah or one's mother. The Radvaz says that Hirhur applies to one's wife. I am astounded. Rashi says that it is merely disgraceful. See Otzar ha'Poskim on 23:3, 8:4. (Ezer mi'Kodesh permits Hirhur about one's wife, even when she is Nidah. It is only Midas Chasidus to refrain. Igros Moshe holds that the Isur of Hirhur during the day that can lead to emissions at night applies even to one's wife.)

Tzitz Eliezer: Alternatively, initially women stayed at home, so seeing a woman outside led to Hirhur. Nowadays women are often outside, so there is less Hirhur. The Levush suggests that this is why people bless sheha'Simchah bi'M'ono even [at a Sheva Berachos] where men and women see each other.

Tzitz Eliezer (ibid., citing ha'Gaon R. S. Z. Auerbach, Ztz"l): One may rely on Leket Yosher in pressed circumstances. When one is walking towards a woman, he is embarrassed to look at her, for she will see this. We are concerned when he is in back, and she does not see him. Initially, women stayed at home, so one could avoid walking in back of women. Nowadays, if one flees from in back of a woman, he will be in back of another woman. It seems that there are more women in the street than men! Unfortunately, Pritzus is rampant today, and men look at women's faces. Therefore, for the sake of a Mitzvah [or if needed for manners], one need not be stringent nowadays.

Suggestion: Nowadays, even Bnei Torah let their wives go first, and Chachamim do not permit. This shows that it is permitted!

Rejection (Mishneh Halachos 5:226): Surely, Chachamim forbade at all times. The Radvaz forbids walking in back of one's wife. People who are lenient are not Bnei Torah. A Kosher Bas Yisrael does not want to make her husband sin. One should not stroll with his wife on Shabbos, due to Bitul Zeman, and acting like an Am ha'Aretz.

Mishneh Halachos (5:227): In big cities, it is almost impossible to avoid going within four Amos of women. The Radvaz forbids within four Amos of a Zonah. One may go within four Amos of other women, but he may not walk in back of them. If there is no alternative, he looks only at the ground. One may do so for Mitzvos, e.g. to go to pray or to work, but not to stroll. If one can go when women are not out, he must, especially in summer, when many women are exposed.

Doleh u'Mashke (p. 290-291): One may not walk in back close enough to see. It is forbidden only if he is directly in back. It is proper not to stand in back of a woman walking, or stand in back of a woman standing, or to [walk in back and] tilt to the side. One may sit in back of a woman sitting, e.g. on a bus. One should be careful about walking in back even nowadays that women are commonly outside, but one need not forfeit his life rather than transgress. One should be careful regarding girls from the age at which men think about them. (Elsewhere, ha'Gaon R. C. Kanievsky Shlita said that this is about six or seven - PF.) One should be careful not to walk in back of one's wife. It is good to be careful not to walk in back of one's daughter.