1. Lots of disruption. If North America is wiped out in nuclear fire, or if Yellowstone erupts, but the rest of the world doesn't collapse all the way back to the Stone Age or worse, and there's a religious revival and a monastic revival as a result, then maybe you get a fair number of monasteries in the Americas because of the immense solitude and distance from everything, and because contemplating the destruction is felt to be a spiritually valuable. Since we're positing almost no population in North America, the monks could easily be 10% or more of the total.
2. Alternatively, you could posit an alternative development of Catholicism in the 19th and 20th Century that broadens the Catholic definition of monasticism to include lay orders like the Opus Dei, the Legionaires of Christ, the People of Praise, and so on, and puts a lot more emphasis on them so lots more Catholics belong to them. I don't think that gets you to 10% though.
3. Alternative 2 works better if monasticism is less exclusive to Catholicism, but that's a POD that probably goes back further than the ARW. Hmm. Suppose Quebec gets included in the Colonies and becomes part of the US. Suppose also as a result that eventually monasticism ends up having some special legal privileges, either in law or custom or legal precedent or some combination of all of the above. Then you might get a situation where groups like the Amish are considered to be monastic (just as laws that were originally designed to protect the privacy of the confessional ended up applying to clergy of most denominations). Over time, this process of development might get you to 10%.