YOU SAID:

I rode through the desert on a horse with no name

INTO JAPANESE

私は名前のない馬に砂漠を通って行きました

BACK INTO ENGLISH

I went through a desert to an unnamed horse.

INTO JAPANESE

私は砂漠を通って無名の馬に行きました。

BACK INTO ENGLISH

I went to the unknown horse through the desert.

INTO JAPANESE

私は砂漠を通って未知の馬に行きました。

BACK INTO ENGLISH

I went to the unknown horse through the desert.