Apparently these two have nothing in common.

The first appears to be a pleonasm while the second sounds like an oxymoron.
The first was a window dressing for a kind of dictatorship that managed to survive for sometime while the second, if ever attempted, would be so volatile that it would ‘evolve’ almost instantly into a ‘dog eats dog’ situation soon to be followed by the most horrid authoritarianism ever known to man.

But there is something that binds them together.

Both had first appeared in the minds of well intended people who were fed up with and trying to do something about what was going around them.

Socialism, the predecessor of ‘popular democracy’ (a.k.a. communism) had grown as a consequence of the excesses committed by some of the ‘savage capitalists’ during the late XIIIV-th and early XIX-th centuries while libertarianism, the reasonable predecessor of libertarian anarchism, as a reaction to the prevailing statism of the late XX-th and early XXI-st ones.

Let me first explain, briefly, why the concept of ‘popular democracy’ is only apparently pleonastic while in reality this wording covers a sheer impossibility. Then I’ll try to extend my practical experience of living under such a regime into a prediction about what would happen if a group of people would ever have to face a truly anarchic situation.

First things first. Democracy means a situation where everybody can voice their concerns about what is going on and where decisions are made in a collective manner, after anyone who cared to had access to all information pertinent to the decisions that had to be made.
In this, theoretic, context ‘popular’ adds absolutely nothing.
In reality ‘popular’ was a window dressing for ‘the population doesn’t know what’s good for it so we, the communists, have to guide it’. Exactly as Marx had explained in the Communist Manifesto. “The Communists, therefore, are on the one hand, practically, the most advanced and resolute section of the working-class parties of every country, that section which pushes forward all others; on the other hand, theoretically, they have over the great mass of the proletariat the advantage of clearly understanding the line of march, the conditions, and the ultimate general results of the proletarian movement.”

Secondly, but not a iota less important, liberty and anarchy are antithetic terms.
Anarchy is absolutely natural. As natural as water boiling in a kettle. It is impossible to say which drop would ‘burst’ first in a bubble and which would be the last to transition from liquid to gas. OK, if you have a mixture of water and alcohol the latter will boil first and the water later but this would happen only if the sill is heated gradually. If the heat source is too strong, a.k.a. ‘uncontrolled’, the process of distillation becomes ‘anarchic’ and the result is lousy – to say the least. In ‘human terms’ this would translate into a ‘dog eats dog’ situation where things become very quickly aligned along a uni-dimensional criterion – usually ‘brute’ force used in a most callous way.
By the way, this is a second ‘connection’ between these two concepts.
‘Popular democracies’, and dictatorships in general, are eroded by the same dissolving force that would cause any anarchic situation to implode – the most callous and less principled members of the group eventually gain absolute control over the rest, only the process takes longer in a dictatorship.

My point being that dictators are constantly being challenged. Both from within and from outside. It is seldom that a dictatorship passes through all its fazes – like Romania’s communist regime did, or Cambodia’s. Usually at some point a group of people understands what’s going on and try to do something about it. For instance what happened in Russia during Perestroika.
Yes, that could have had better results but just imagine Russia going down to the same depths Romania has probed almost 30 years ago. When most (non)public offices were held by incompetent nincompoops whose only goal was to prolong their survival by serving their demented master. Could you have slept at night knowing that Russia’s nuclear arsenal was being managed by such idiots?

Most dictatorships are being ‘weighted down’ by tradition and cultural norms. A dictator needs some time before they can do what they want and they can almost never accomplish all that they would like to. Good or bad.

On the contrary, in a truly anarchic situation – when no rules are observed anymore, except for ‘he who has the biggest fist prevails’, of course – things degenerate very fast. And need a lot more time to get back on track.
Like what happened during the French Revolution. When “the Revolutionary government (the ‘big fisted’ guys of the moment) decided to make “Terror” the order of the day (September 5 decree) and to take harsh measures against those suspected of being enemies of the Revolution (nobles, priests, hoarders). In Paris a wave of executions followed. In the provinces, representatives on mission and surveillance committees instituted local terrors. The Terror had an economic side embodied in the Maximum, a price-control measure demanded by the lower classes of Paris, and a religious side that was embodied in the program of de-Christianization pursued by the followers of Jacques Hébert.”

You might be wondering how come that such a generous concept like ‘let’s treat the workers fairly’ was high jacked into the horrors of communism and whether the same rationale could be extended to predict what a libertarian-anarchist society might (d)evolve into.
The way I see it people’s imagination is huge. A lot of things that might seem bland to ordinary eyes are perceived as resources by ‘crafty’ people and a lot of situations that seem helpless, or even desperate, to normal human beings are seen as very good opportunities by those adept at fishing in troubled waters.

It’s exactly the individuals where this kind of ‘craftsmanship’ is associated with ‘moral lassitude’ who would spare no effort in their attempt to make the’ best’ of the opportunities present in a country being run in an authoritarian manner or during an anarchic situation.

For this kind of guys it doesn’t matter whether the ancient regime was toppled by some socialist utopian (for instance Kerensky in Russia or Dr. Sun Yat-sen in China) or by a bona fide dictator (like, for example, Jorge Rafael Videla of Argentina under whose regime some 10 000 to 30 000 people have been ‘disappeared’ by the authorities). Or whether the anarchic situation has been a consequence of regimes imploding from within (pre-revolutionary France, communist Russia, yesterday day Libya) or being unsettled by  sloppy outside interventions  (Afghanistan and Iraq)

All these situations, and many others, are the perfect breeding and hunting ground for  ‘political hyenas’, callous ‘operatives who would eventually ‘denature ‘even the most well intended dictatorship or ‘well organized’ libertarian anarchy.

I’ve already experienced one of this situation.

I really don’t want to experience the other. No matter how appealing it might seem to the libertarian ‘fundamentalists’.