You can be a Jew and recognize Jesus as being real. I recognize Jesus as a literal person who existed. I say recognize because Jesus existing is not a fact, we don't have enough extra biblical evidence to prove Jesus existed in the way we can prove that Pontius Pilate existed. I just believe Jesus existed because of the dedication his followers showed in spreading the teachings of Jesus and were willing to die for these teachings, I can't imagine doing that for someone that never existed.
But in terms of can you be Jewish and believe in Jesus as Messiah... That's almost impossible to do, for many few reasons.
The Messiah was supposed to bring world peace. I believe Christians as a group have killed more people than anyone else. Peace came no closer to reality because of Jesus.
The Messiah was supposed to help the tribes of Israel first, and through the nations of Israel reaching such a high point of righteousness, the rest of the world would be enriched. Even Jesus himself claimed he was here only for the lost sheep of Israel. Unfortunately within his lifetime Jesus did not help the 12 tribes of Israel in any meaningful way. Even though his disciples apparently accomplished a lot more, it did not bring about a positive change for the 12 tribes of Israel.
It can be argued that Jesus had no intention of starting a new religion, and I believe this is true based on my readings of the gospels and on historical realities of early christians before the centralization of the catholic church. Early church communities such as the Ebionites are known from Catholic records as having Hebrew copies of Matthew that taught that Jesus was a man, the son of Joseph, and that all "Christians" need to be observant, and possibly even Pharisaical Jews. As a result of these very Jewish beliefs, they were destroyed by the Catholic church. Fast forward thousands of years to all major sects of modern Christianity, we find you MUST believe Jesus was not the son of Joseph, you MUST believe Jesus did away with the laws of Israel/God, and you MUST believe that Jesus is somehow also God? All of those ideas are impossible to have in Judaism.
Continuing with that line of thinking, if you want to "be a Christian," you are required to have beliefs that are incompatible with Judaism. If you want to believe in Jesus in a real legitimate Jewish way like the Ebionites, then you are not a Christian. If you want to believe that Jesus was a Rabbi, one of many Rabbis that have existed, you could possibly remain a religious Jew, but Christians would reject you. The only way to be a part of a modern Christian movement is that you are required to have beliefs that are just incompatible with Judaism. Even organizations like Jews for Jesus preach things that are incompatible with Judaism.
Is it possible to believe in a figure named Jesus and still be religiously Jewish? Sure I think it's possible, but so unlikely that it's almost impossible. Because you would have to reject 99% of Christian dogma, teachings, and beliefs about Jesus AND you'd have to reject all Christian scriptures outside of the gospels, and also exclude Luke from being part of the gospels.
For those who want to know about historical sources for Jesus. We have a reliable source by the Roman Senator and Historian Tacitus. You can read the background of the text in the source but I'll post the contents of the source below. I'm posting it to show how possible it is that Jesus existed, and to illustrate how little we know of the actual historical Jesus. Most of the best sources are really only discussing the current existence of the followers of Jesus.
But all human efforts, all the lavish gifts of the emperor, and the
propitiations of the gods, did not banish the sinister belief that the
conflagration was the result of an order. Consequently, to get rid of
the report, Nero fastened the guilt and inflicted the most exquisite
tortures on a class hated for their abominations, called Christians by
the populace. Christus, from whom the name had its origin, suffered
the extreme penalty during the reign of Tiberius at the hands of one
of our procurators, Pontius Pilatus, and a most mischievous
superstition, thus checked for the moment, again broke out not only in
Judæa, the first source of the evil, but even in Rome, where all
things hideous and shameful from every part of the world find their
centre and become popular. Accordingly, an arrest was first made of
all who pleaded guilty; then, upon their information, an immense
multitude was convicted, not so much of the crime of firing the city,
as of hatred against mankind.