It is explained here that a Jew cannot lose his 'Jewishness', It seems from this answer that Jewishness is a genealogical status and little to do with religion. So is it possible for a Jew to practice Islam and Judaism simultaneously? Are there any examples of such? One example that comes to my mind is about the famed Sabbatai Tzevi who was a Jewish scholar who became Muslim and continued to show both Jewish and Islamic behaviors; are there any more examples?
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The two questions that are nearly duplicates of this discuss whether a person is considered from the perspective of Halachah to be Jewish if he converts to another religion, and what the person's status is generally. But they don't, IMO, really address this specific problem.
Here's the short answer: No.
Here's the more complex answer: No. Unless you mean is it physically possible for a Jew to practice another religion, in which case, obviously, a person has free will to do what he pleases. But he will be sinning. You see, just because Judaism does not acknowledge a person's conversion to another religion (ie., Judaism views that person as being a repeat and regular sinner and violator of Jewish law), that does not mean there is any particular recourse.
Shabetai Tzevi is universally regarded by Jews the world over as a heretic and sinner. He also, incidentally, regarded himself as the Messiah, and most of his followers did too, until he started acting all weird - converting to Islam being the final straw for the few who still hung on to hope that the Messiah had arrived.
In other words, sure, a person could be a Jew who totally violates every Jewish precept, whether that be because the person just doesn't care or because the person converted to another religion, and yes, that person would, technically, still be considered a Jew. But no, such practice is not allowed, sanctioned, or tolerated, and certainly not recognized. No "dual-citizenship" here.