The prohibition of "Bishul Akum" forbids partaking of food cooked by a gentile. Even if all the ingredients are perfectly Kosher, the food is forbidden by virtue of the fact that it was cooked by a non-Jew. Sephardic custom is more stringent than Ashkenazic practice with regard to this Halacha. Ashkenazim allow eating food cooked by a gentile if a Jew participated in any capacity at any point in the cooking process, even if he simply kindled the flame or turned on the oven. Sepharadim, however, forbid food cooked by a non-Jew unless a Jew played some active role in the process of cooking, such as by pouring the ingredients into the pot or mixing the food.
The question arises as to whether this Halacha will affect the status of canned tuna fish, which undergoes a cooking process before it is canned. In Israel, it is likely that the ones pouring the fish into the steamer are Jewish, or that a special Mashgiah (supervisor) is hired to do so, so that all the fish is cooked by a Jew. In the United States, however, it is unrealistic to expect that only Jews pour the fish into the steamer at the factory. Some brands of tuna are marked as "Bishul Yisrael," stating that they were cooked by a Jew, but this means only that a Jew turned on the steamer. As mentioned, this suffices to avoid the prohibition of "Bishul Akum" only for Ashkenazim, but not Sepharadim.
Some allowed eating tuna fish in light of the fact that the "Bishul Akum" prohibition applies only to food that is not customarily eaten raw. Sushi, or raw tuna, is a delicacy in the Far East, and we might therefore claim that tuna is not subject to the prohibition of "Bishul Akum." It would seem, however, that since sushi is not so common here in the United States, we cannot classify tuna as a food that is customarily eaten raw, and thus we cannot rely on this argument.
In any event, Hacham Ovadia Yosef ruled that canned tuna does not fall under the prohibition of "Bishul Akum" for an entirely different reason, namely, that Halacha does not equate smoking with cooking in this respect. "Bishul Akum" does not apply to foods that were smoked or steamed by a gentile, and thus tuna is not subject to "Bishul Akum" despite the fact that it is steamed during its production process. Hacham Ovadia notes that a number of earlier authorities held this view, including the "Darcheh Teshuba" and "Serideh Esh." Although others disagreed, we may rely on this lenient position to allow the consumption of tuna fish.
Other suggested a different basis for allowing tuna fish, namely, the Halacha that restricts "Bishul Akum" to foods that are worthy of being served to noblemen and dignitaries ("Oleh Al Shulhan Melachim"). Fresh tuna might be served to kings and princes, these authorities claimed, but canned tuna is not respectable enough to be brought before men of royalty. For this reason, too, the prohibition of "Bishul Akum" should perhaps not apply to tuna.
It should be noted that other complex issues are involved regarding the Halachic status of tuna fish. Our discussion here related only to the particular issue of "Bishul Akum," and we concluded that this prohibition does not apply to tuna fish. Other important issues, however, must also be considered in determining the permissibility of tuna fish.
Summary: The prohibition of "Bishul Akum," which forbids partaking of food cooked by a non-Jew, does not apply to smoking or steaming, and, as such, this prohibition does not apply to tuna fish.
