Greetings, Mr Anonymous. Before we can answer this question, the answers to a few smaller pre-questions must first be clarified:
What exactly is a "real Jew"?
How do you determine who is a "real Jew"?
Are there any limitations on accuracy in determining who is a "real Jew"?
How do you determine a "fake Jew"?
According to Judaism itself, a Jew is someone who was born to a Jewish mother or who converted successfully. There are no "real" or "fake" Jews; someone is either Jewish or they're not.
Let us assume that by "real Jew" you refer to someome descended from the original Israelites of the Bible. Already we have a problem; this is not quite how a Jew is defined. A notable number of non-Jewish Americans have some Jewish ancestry, but this doesn't make them "real Jews". The Israelites were an ancient people of whom only skeletons now remain; how do we genetically determine who is descended from them? We can compare genetic ties between local groups who are likely to have been descended from related ancient peoples, but this isn't a direct insight into Israelite genealogy and so comes with uncertainty. On the historical front, how limited are we in determining whether or not a modern group can trace itself back to an ancient group? Even just a few hundred Israelites could have started what today is the Ashkenazi Jewish population; how sure can we be that historical evidence around their migration to Europe even exists? Any conclusions we reach by genetic and historical evidence will have fundemental limitations that bring about uncertainty.
The best historical and genetic evidence would suggest that some, but not all of the Palestinian population, is descended in some way from the ancient Israelites. The same however also applies to Ashkenazi Jews, Sephardic Jews, Samaritans and a number of other groups. They are all "real Jews" according to our assumed definition, however by Torah law only the modern day Jews can call themselves Jews at all. As already discussed, in Judaism a person is either Jewish or not Jewish, not "real" or "fake".
I will conclude this post with a couple of rhetorical questions:
1. In what way does the ancient ancestry of someone make a difference to whether or not they should be allowed to remain on land they legally own?
2. In what way does ancient ancestry automatically make someone's actions right or wrong?
Thank you for your contribution.