Yes. When persistent sin is public knowledge, the priest should withhold the Eucharist also from a Catholic. When the sin is unknown, the presumption is that the Catholic confessed it and received absolution. When a known non-Catholic approaches to receive, the presumption of being in the state of grace cannot be made, and the Eucharist is withheld. That is because at least one thing that is known about the person is that he or she resisted conversion so far.
You then go on to say that you recognize the Real Presence of Christ. So then you are in a curious position where Christ asked you to eat of His body, you know that He did, yet you won't do it for one reason or another. This is hard to reconcile with freedom from sin, and hence with the conditions spelled out in 1 Cor 11.
There are Christians who are not Catholic.
But the natural man receiveth not the things of the Spirit of God: for they are foolishness unto him: neither can he know [them], because they are spiritually discerned. - I Cor 2:13-14
For that reason I would not say that every communion in a particular assembly or location is divine. Nor is it exclusive. God makes it so - or not - whether Catholic, Lutheran, Orthodox, Baptist, the bread and wine as a Christian family observes Shabbat, etc.
Other examples from Scripture would be God's presence in the burning bush, the pillar of fire and smoke, the Temple. And of course, most especially, in my brothers and sisters in Christ:
So when you say "Real Presence" we may not be talking about the same thing even though the words would fit.
xone provided a very illuminating pdf about close communion in the Lutheran doctrine. If you have a link for a similar Q&A on the Catholic doctrine, I'd like to read it. Without further information, I suspect the Catholic doctrine claims a Spiritual discernment as well as a physical discernment in its exclusion of non-Catholics.
God's Name is I AM