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From the verse, “Consecrate to Me every firstborn, whatever is the first to open every womb among the children of Israel, both of man and of animal; it is Mine” (Exodus 13:2), the Sages derived that the law of the firstborn animal applies only to an animal owned by an Israelite and not to an animal owned by a gentile.
The Tannaim disagreed regarding the status of an animal jointly owned by an Israelite and a gentile: according to Rabbi Yehudah, such an animal is subject to the law of the firstborn, whereas according to the Sages, only a firstborn animal that is wholly owned by an Israelite is subject to this obligation.
The more interesting development comes with Tosafot. The Gemara appears uncomfortable with intentionally selling a piece of the fetus to a gentile simply to escape kedushat bekhor.
We explore ha ‘aramah, engineered around a halakhic category, yet its purpose is not merely convenience. It prevents a more serious halakhic failure. The artificiality of the mechanism is tolerated because the underlying concern is genuine.