In the United States, while it is extremely rare, it is possible to be summoned for Federal and State jury duty on the same day as depicted in this picture taken from a Reddit post:
When this happens, which jury duty must the summoned attend?
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I don't know that there's any law explicitly answering this, but it seems that the answer would have to be that the state court summons must yield to the federal court summons.
Under the Supremacy Clause, any state law that makes it impossible to comply with a federal law is preempted and cannot be enforced in those circumstances. Penn. R. Co. v. Ills. Brick Co., 297 U.S. 447, 459 (1936) (“To the extent, if at all, that there is conflict between state and federal regulation, the latter must prevail.”).
Because enforcing the state summons would make it impossible to comply with the federal summons, the Supremacy Clause would prohibit the state from enforcing it, though it could probably still require the recipient to notify the state court of the conduct and make himself available at a future date.