This is basically a question on a Mishna Sanhedrin 9, 6, and is concerning difference between Jewish men and women concerning the prohibition for relationship with non-Jewish partner.
I will divide the topic in 3 (based on Baal Hamaor and Ramban Rabenu Yona and Ran for the moment):
- A man with a non-Jewish woman;
- Ramban in Milchamot stated that it is similar to death penalty or cut off because avengers are allowed to kill him. There is also a din of cut off, learned from the verse in prophets (Mal'achi 2, 12) in Gemara Sanhedrin 82a.
A non married Jewish woman with a non-Jewish man;
for a Jewish woman who have a relationship with a non-Jew, there is no law of "avengers are allowed to kill her"(1). It seems from the Yerushalmi (cited in Mel'echet Shlomo here that the reason for cut off is the consequence, birth of enemies of Israel, Ramban in Milchamot explain that the inverted situation, a JW with a NJM leads to birth of a Jewish son which will not be enemy of Israel. So no cut off and no avengers against a JW in relationship with a NJM).
But the Ran at the beginning of the paragraph in Chidushin 74b Sanhedrin think that there is no gender discrepancy and a Jewish woman (even not married) in relationship with a non-Jew man in relationship is included inside the rule of "the avengers ...". (See here at the right top). __ See also Mel'echet Shlomo Mishnayot Sanhedrin 9, 6 here in name of Tur CM 425 in name of Shilte Hagiborim Perek Hacholets on Mordechai, in name of Rav Avraham Hagadol the same opinion.
A married Jewish woman with a non-Jewish man.
Following the Baal Hamaor in Sanhedrin Gilui Arayot as married woman with a Jewish man. He said that Esther (with Achashverosh) was Gilui Arayot because of her marriage to Mordechay. But he consider further that it may be different from Eshet Ish because of the verse
"וְאִישׁ אֲשֶׁר יִנְאַף אֶת אֵשֶׁת אִישׁ אֲשֶׁר יִנְאַף אֶת אֵשֶׁת רֵעֵהוּ מוֹת יוּמַת הַנֹּאֵף וְהַנֹּאָפֶת."
in which אשת רעהו indicates that both, husband and lover are Jews (Rabenu Yona rejected this proof from the verse and said that this verse excludes Eshet Goy only). But his opinion is as described above. Ramban said that French Scholars also think that Ester was Giluy Arayot (and Rava stated that the allowance despite Arayot and Farhesia was because a conjonction of Karka Olam (for Arayot) and Hanaat Atsman (for Farhesia)). The Rav Avraham Bar David also thinks as them. The Rabenu Yona in Chidushim cited Gemara Sanhedrin 57b that a Non-Jew who has a relationship with Ervat Israel(2) is penalized by strangulation. He concludes that he is killed, she is also killed.
- The Ramban in Sefer Milchamot Said that with a non-Jew it is not similar to Naara Meurasa (no death penalty, and no cut off(3)) so, not Yehareg veal yaavor. a married Jewish woman who have a relationship with a non-Jew, there is no law of "avengers are allowed to kill her".
(1)Ramban explained in Milchamot that for a Jewish man who has a relationship, considering that avengers are allowed to kill him, it is equivalent to death or cut off penalty (מסתברא דישראל הבא על הנכרית מיקרי גילוי עריות משום דקנאים פוגעים בו והוה ליה כחייבי מיתות דשייך בהו מיתה וכרת כדאיתא לקמן במכילתין)
(2) Rashi: בעריות דידן. באשת איש ישראל:
(3)והא דאמרינן גבי אסתר דלהנאת עצמן מותר משום דלאו גילוי עריות הוא וכדאקשינן בגמרא והא אסתר פרהסיא הואי ולא אקשי גילוי עריות הואי משום דלא מקרי גילוי עריות אלא בנערה המאורסה שזינתה אי נמי אשת איש אבל נכרי הבא על בת ישראל לא מקרי גילוי עריות דלאו מיתות ולאו חייבי כריתות נינהו לא אתיא מנערה המאורסה דקרא אלא חייבי לאוין [נינהו] כדמפורש בשמעתא לעיל.