The pillars of the Jewish religion
1. Keeping the Sabbath and the ten commandments
2. Rules concerning what is clean and what isn't. (Kosher food, etc)
3. Rules concerning your covenant with God, including circumcision.
Anyone claiming to be "orthodox" must address these three pillars so let's look at that.
1. There are four sabbaths per month, the first quarter of the moon, the full moon, the third quarter and the new moon which can be 2-3 days. Any Jew who is on the Gregorian or Julian calendar is not keeping the Sabbath. The Old Testament clearly and emphatically condemns keeping the sabbath on an "unholy" day. Imagine keeping the Sabbath on a day named after Satan, the enemy of God. "Saturday" is not the sabbath day. God is sovereign, He rules on the throne, if the world does not allow you to keep the Sabbath you need to ask God why that is. Claiming you can keep the sabbath on "Saturday" is like trying to clothe yourself with a fig leaf. There are many other laws like giving the land a sabbath rest every seven years as well. Only in Israel could there be anyone who comes close to this, but even there many make excuses for working on sabbath days.
2. Kosher doesn't simply mean you don't eat pork. You can't keep these laws if you eat at a restaurant or home that cooks pork. There are many other foods that are unclean as well. No "orthodox Jew" can eat at Mcdonald's or pretty much any restaurant worldwide. It is very difficult to live according to these laws while not living in Israel. This includes the break room at the office. So again, it is almost impossible for someone who is not in Israel to keep these laws. Again, claiming to be "orthodox Jew" while not keeping these Kosher laws is insulting to God.
3. Circumcision indicates they have a covenant with God, it is a covenant relationship. Like a wedding ring. This covenant relationship includes the land of Israel. Imagine you were to marry a man, but you didn't want to live in his house or even in the same country where He is living. It is absurd.
Therefore I would argue that only Christians can possibly be "orthodox Jews". Jesus is the Lord of the Sabbath, if He decides to change the day we remember Him to the Lord's day He has that right. Jesus on the cross cleansed all food and all people. So because of the cross of Christ we can go everywhere and bring the gospel to everyone. The cross of Christ is our sign of our covenant. If our heart has the marks of the cross on it, if we have a new heart, then we can dwell with Jesus in our heart everywhere. We don't need to worship on a mountain in Israel, we can worship the Lord everywhere because the church is the house of God.