Why "market socialism" is not socialism?

I looked but couldn't find anything in the library that argues for why 'market socialism' is not really socialism. So maybe we can use this thread to encourage someone to make an article or at least have arguments at hand.

Market socialism is really popular these days. Like in reddit forums, people make arguments in favor of it all the time. There are some who even call themselves "libertarian market socialists," so it's even popular among anarchist types. I think it's a tendency that holds sway among people who don't really have a proper critique of capitalism and a full understanding of socialism.

Market socialists often define socialism as simply workers' control of the means of production. So you can see how they fit it in under the umbrella of socialism. Like for them, it doesn't matter that you still have businesses competing in the marketplace, all that matters is that those businesses are owned and controlled by the workers who work in them. It doesn't strike them as odd that in a supposedly socialist society, people are fragmented and competition is the main driver of economic and social life.

I think this is why it's important to define socialism is the ownership of wealth by and for the benefit of society as a whole. Ok, so how would you argue against market socialism, if your against it?