In one sense, we already have global anarchy. I mean, there isn’t one consolidated world government, is there? Then again, we live in a global community, so there is a world government of a sort. But, to answer your question seriously, no, global anarchy is not on the horizon. Not now and probably not ever.
All of the demonstrations and radicalism you see on TV are par for the course. Such things have been part and parcel of nation states since their inception. (History textbooks might be instructive here.) Hell, even ancient monarchies suffered from insurrections. The Spartans were so afraid of slave rebellions that they declared war on the slaves once a year, just to keep them in line.
Modern agitation seems stark only because we live in a technological age, in the age of the 24-hour news cycle. I think it’s safe to say most of the protesters want to reform their governments, not abolish them.
Anarchy is an individual rebellion against forms, not a social utopian construct. Bottom line: Individual anarchy has already arrived. In fact, it’s always been around as the province of a minority of vagabonds. Social anarchy on the other hand, is a non-existent Elysium, and will only ever be the figment of optimistic imaginations.
Don’t wait around for that shit. It’s a waste of time. I forget who said it, but I love this quote: “I don’t need to wait for the Übermensch. I AM the Übermensch.”