Why there's no Mishnah Bavli?

Question

The question might seem odd, I admit, but
In the beginning pages of the Bavli Sanhedrin (5a), it states that the power of the ראש גלותא in Babylon was bigger than that of נשיא in the land of Israel in the time of R' Yehuda Hanassi (see the whole Suggyah of certifying judges).

"פשיטא מהכא להכא ומהתם להתם מהני ומהכא להתם נמי מהני דהכא שבט והתם מחוקק "

How come R' Yehuda Hanassi in the Holy Land decided on the codification of the Mishnah, but no ראש גלותא did so also, for they really could - they had the brains and the political power to force their decisions on all others Rabbis? Why was the "Israeli Mishnah" accepted unanimously even though the Israeli Rabbis were not trusted so much in Babylon?

Bonus wonder: how would the Bavli Mishnah be different from R' Yehudah's?

Answer 1

I think the simplest reason why is because of the influence of Rav and Shemu'el. Most of the Babylonian amora'im can be traced back to them somehow through their students. Having learned personally from Rabbi Yehuda Hannasi, they would have wanted to continue teaching as they learned, through the mishnayot of the land of Israel. Their influence is clear from the fact that later generations were apparently more familiar with their disputes than with certain parts of the Mishnah itself (Berachot 20a). Pesachim 30a is another case in which Shemu'el is influential in an actual case deciding the law for pot-sellers. The fact that Rav and Shemu'el, unlike the Resh Galuta, were actually going and teaching people, is probably the biggest reason for the Mishnah, particularly that of Rabbi Yehuda Hannasi, to have become the code of law for Babylonian Jews.

Regarding the question of how the Babylonian Mishnah would be different from the Mishnah we have: The Mishnah already leans Babylonian. The Mishnah already has a strong preference for the opinions of the school of Hillel the Babylonian (as he is called in Pesachim 66a) over those of the school of Shammai (see for example Berachot 1:3). (This is probably another reason why it wouldn't have been hard for Babylonians to accept it.)

There are Babylonian rabbis who aren't quoted in the in the Mishnah we have, but whom we do know from baraytot (e.g. Rabbi Yehuda ben Betera, who lived in Nisibis according to Pesachim 3b). I don't know whether they had a set of teachings based on something like the Mishnah Rishonah (Sanhedrin 3:4) that Rabbi Akiva or Rabbi Yehuda Hannasi worked from, or whether they had only the unconnected teachings that we have attributed to them in baraytot. A Babylonian version of the Mishnah would probably have included more of these people as tanna'im. In any case, these tanna'im did make it into the final law by way of the baraytot, so they weren't forgotten despite not having made it into the Mishnah.

Answer 2

Because in Issur VeHetter, the Sanhedrin in Eretz Yisrael had authority over the Reish Galusa.

The Gemara says in Pesachim 51a

ורבה בר בר חנה לית ליה הא דתנן נותנין עליו חומרי המקום שיצא משם וחומרי המקום שהלך לשם אמר אביי הני מילי מבבל לבבל ומארץ ישראל לא"י אי נמי מבבל לא"י אבל מא"י לבבל לא כיון דאנן כייפינן להו עבדינן כוותייהו

The Gemara asks: And is Rabba bar bar Ḥana, who was lenient with regard to a matter that is prohibited, not in agreement with that which we learned in the mishna: When one travels from one place to another, the Sages impose upon him the stringencies of the place from which he left and the stringencies of the place to which he went? Abaye said: That applies when one travels from one place in Babylonia to another place in Babylonia, or from one place in Eretz Yisrael to another place in Eretz Yisrael, or alternatively, from Babylonia to Eretz Yisrael. However, when traveling from Eretz Yisrael to Babylonia, no, this principle does not apply. Since we, the residents of Babylonia, are subordinate to them in terms of halakha, we act in accordance with their custom, but a resident of Eretz Yisrael is not required to follow the Babylonian custom.

(Emphasis mine).

The reason is, as Rashi says:

אנן כייפינן לבני א"י דאינהו סמיכי ובבבל לא סמיכי

We're subject to those in Israel, since they have Smicha and we don't.

So the answer is that the Beis Din in Israel has authority since it has Smicha.

Tosfos says another answer:

בני א"י חכימי טפי דאוירא דא"י מחכים כדאמר הכא דאיקרו מחוקק שמלמדין תורה ברבים The Torah scholars in Israel are wiser, since the air of Israel makes one wise, as the verse says "staff" - they teach Torah in public.

So why does the Reish Galusa have primacy in appointing judges?

Rashi says (on the Gemara you mentioned):

לשון שררה ויש להן רשות להפקיר דהפקר ב"ד הפקר דכתיב (עזרא י) וכל אשר לא יבא וגו' ביבמות בהאשה רבה

They have permission to declare objects owner-less, since the court has the power to make objects owner-less.