Midrashic sources regarding the Sea splitting into twelve paths

Question

Tehilim 136:13 says:

לְגֹזֵר יַם סוּף לִגְזָרִים כִּי לְעוֹלָם חַסְדּוֹ

To Him Who cut the Sea of Reeds asunder, for His kindness is eternal.

Rashi's commentary indicates that this is talking about God cutting twelve paths through the sea, one for each tribe. Does anyone know what Rashi's source in the Midrash is, and where this idea shows up for the first time in the Midrash?

This question was adapted from a Facebook post by DovBear.

Answer 1

Toras Shleimah (15:127*) brings the Midrash of the Yam Suf splitting into 12 lanes for the twelve tribes in the name of two of the "smaller midrashim": Midrash Vayosha (Chapter 12) and Midrash Temurah (Temurah 13).

Answer 2

Pirkei d'Rebbi Eliezer ch. 42 and Midrash Shocher Tov chapter 114,see also the Rav on Avos 5:3.

Answer 3

After quoting the Mechilta on Beshalach that says that the Sea was split into parts, HaTorah HaTemimah brings 3 sources that says it was split into 12:

• Midrash Shocher Tov 114

• Targum Yonatan (I'm assuming on 136:13)

• Rambam's Commentary on the Mishnayot, Avot 5:4

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I wasn't able to find where the Midrash Shocher Tov mentions 12. (What I think is) the relevant passage starts on the bottom of this page, but it does not bring a number.

The notes by MaHaR"i HaKohen do mention 12 parts. He says this could be hinted to by the fact that the verse says, "The water was to them as a wall". "to them" appears to be extra, but comes to teach us that each tribe had a wall, hence 12 parts.