Non-Jewish courts are referred to as arka'ot (ערכאות), although the precise etymology of this word is debated, and as agoriyot (אגוריות). There is a longstanding prohibition against having one's case heard in one, although nowhere does this prohibition have anything to do with shituf. The oldest source is the Talmud Bavli:
ר"ט אומר כל מקום שאתה מוצא אגוריאות של עובדי כוכבים אע"פ שדיניהם כדיני
ישראל אי אתה רשאי להיזקק להם שנאמר ואלה המשפטים אשר תשים לפניהם לפניהם
ולא לפני עובדי כוכבים דבר אחר לפניהם ולא לפני הדיוטות
Rabbi Tarfon says, "Any place that you find non-Jewish courts
(agoriyot), even though their laws be the same as the laws of Israel,
you are not allowed to resort to them, since it says "these are the
laws that you shall set before them" (Exodus 21:1). Before them
and not before non-Jews. Alternatively, before them and not
before lay tribunals.
• Babylonian Talmud, Gittin 88b
This passage is quoted in the Sheiltot of R' Ahai Gaon (שאילתא ב', בראשית), although it adds the stipulation that this is even if the non-Jewish courts are operating in accordance with biblical law (אע"ג דקא דיינין דינא דאורייתא), and it gives the ruling in the name of Rabbi Meir instead of Rabbi Tarfon.
The same prohibition is to be found twice in Midrash Tanchuma: Mishpatim §3 (where it is presented anonymously) and §6 (where it is given in the name of Rabbi Shim'on, but in which it is otherwise identical to the passage in bGittin 88b). The first of those two instances is especially interesting, since it continues as follows:
שכל מי שמניח דייני ישראל והולך לפני אמות העולם כפר בהקדוש ברוך הוא
תחלה ואחרי כן כפר בתורה
All who abandon Jewish judges and who present their case before
non-Jews are guilty at first of denying God, then later of denying the
Torah.
• Tanchuma, Mishpatim §3
The passage then continues with an allegory: a doctor visits one sick man, and tells him that he may eat and drink what he pleases. He visits another, and gives him a specific diet. When questioned by the second man's family, he tells them that the first man is going to die, so what he eats and drinks is of no consequence. Likewise, the laws of the nations are of nothingness, while the stipulations of Israel lead one to the paths of life. The passage concludes:
אמר הקדוש ברוך הוא לישראל אם עשיתם אתם את הדין ואין אתם מזדקקין לפני
אמות העולם אבנה לכם בית המקדש וישבו בה סנהדרין
God tells Israel, "If you render judgments for yourselves and do not
resort to non-Jewish courts, I will build a temple for you and the
Sanhedrin will sit within it"
• Tanchuma, ibid.
If you consult the mediaeval mefarshim on Exodus 21:1, you will find the same ruling (quoted from the Tanchuma and from Gittin 88b) in the commentaries of both Rashi and Ramban. Rashi's wording is of particular interest:
לפניהם ולא לפני עובדי כוכבים ומזלות ואפילו ידעת בדין אחד שהם דנין אותו
כדיני ישראל אל תביאהו בערכאות שלהם שהמביא דיני ישראל לפני עובדי כוכבים
ומזלות מחלל את השם ומיקר את שם האלילים להשביחם
Before them, but not before non-Jews. And even if you know that they
are operating with the same law that the Jews employ, do not bring him
to their courts (arka'ot), for whoever brings Jewish cases before
non-Jews desecrates the name of God, and elevates foreign gods with
praise.
• Rashi, Exodus 21:1
As halakha, this ruling is to be found in the Rambam as follows:
כל הדן בדיני גוים ובערכאות שלהם אע"פ שהיו דיניהם כדיני ישראל הרי זה
רשע וכאלו חרף והרים יד בתורת משה רבינו שנ' ואלה המשפטים אשר תשים
לפניהם לפניהם ולא לפני גוים לפניהם ולא לפני הדיוטות
All those who try their cases in accordance with non-Jewish law or in
non-Jewish courts (arka'ot), even if their laws are as the laws of
Israel, is a wicked person and it is as though he insulted and struck
the Torah of Moses, since it says "These are the laws that you should
place before them". Before them, and not before non-Jews. Before
them, and not before lay tribunals.
- Rambam, Hilkhot Sanhedrin 26:7
Likewise, we find this ruling in the Tur and the Shulchan Arukh:
אסור לדון בפני דייני עכו"ם ובערכאות שלהם אפילו בדין שדנים כדיני ישראל
ואפילו נתרצו ב' בעלי דינים לדון בפניהם אסור וכל הבא לידון בפניהם הרי
זה רשע וכאלו חירף וגידף והרים יד בתורת מרע"ה
It is forbidden to reach judgment before non-Jewish judges, or in
non-Jewish courts (arka'ot) - even in a matter that they judge like
Jewish judges do. Even if both plaintiffs want to be judged by them,
it is forbidden, and all who come to be judged before them are
accounted wicked, and it is as though they insulted and cursed and
struck the Torah of Moses.
- Shulchan Arukh, Choshen Mishpat 26:1.
See the Rema (ibid.) for a situation in which you are allowed to use non-Jewish courts in order to place pressure upon one of the plaintiffs to desist, and see also the Rambam (ibid.) and the Tur/Shulchan Arukh (op.cit. 26:2) for situations in which non-Jewish courts can be used to place pressure upon Jewish plaintiffs who refuse to appear in a Jewish court.
See also Menachem Elon, Jewish Law: History, Sources, Principles (Vol. I; trans. B. Auerbach and M.J. Sykes; Jerusalem: JPS, 1994), 13-18. He brings various examples of edicts and responsa, permitting (under certain circumstances) the uses of non-Jewish courts to "decide on matters in which the general government had a special interest, such as real estate, the payment of promissory notes, government taxes, currency matters, and assault and battery" (op.cit. 17).
Not all Jewish authorities have accepted the use of such edicts, but amongst those who did are the Maharach (Rabbi Hayyim Eliezer ben Yitzhak, son of the Or Zarua), Nachalat Shiva (Rabbi Shmuel haLevi, 17th c. Poland), the Baal haNetivot (Rabbi Yaakov Lorberbaum, 18th-19th c. Poland), and the Rema. Those who disagree with such edicts include the Ramban, Rabbeinu Tam, the Rashbam, the Rashba, and (apparantly) the Maharam of Rothenburg. One example of a scholar who permits the use of non-Jewish judges, but only under certain circumstances (specifically, if both litigants name a particular non-Jewish judge whom they trust) is the Shakh (Choshen Mishpat 22:15).
