"不排他法律管辖区" can be translated as "non-exclusive jurisdiction" or "jurisdiction without exclusivity" in English.
In legal contexts, "non-exclusive jurisdiction" is a more commonly used term to describe a situation where multiple jurisdictions may have the authority to hear and decide a case, rather than one jurisdiction having exclusive authority. This term emphasizes that the jurisdiction in question is not the only one with the power to adjudicate the matter.
"Jurisdiction without exclusivity" is another way to express the same idea, though it is less commonly used in legal documents and discussions. It directly conveys the absence of exclusivity in the jurisdiction's authority.
For most legal and official purposes, "non-exclusive jurisdiction" would be the preferred translation.