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has? No, they only control it. Anarchist syndicalism is simply the workers taking control of what they have always used and keeping what they produce, instead of allowing it to be stolen by the rich who do no labor themselves but somehow "own" (control) the products the working class creates.

https://www.marxists.org/archive/marx/works/1847/wage-labour/

This explains the nature of the control.

I read the link. I think you assume, what I know about the state, rather than asking me first on what I know. The Marx paper assumes nobody crosses over from being a worker to being a business owner. When I started a business, I discovered that all of my customers were capitalists in regard to how they wanted to use their currency and capital. Nobody wanted to pay more than they could get by with regardless of the benefit to those who worked with me.
Marx definitely brought up reasons to envy, but each person must make choices in how they will live, how they will provide for themselves and what they will value. I value not taking someone's stuff, just because I want it. Just as I value that someone else, doesn't take my stuff, just because they want it.
The Communist states that have formed have led to the deaths of millions. Would Anarchist syndicalism be different? I can not say, however, I am not keen to see the logical consequences of it being put into action, based on past Communist experiments.

"consequences of it being put into action, based on past Communist experiments."

Which part dont you like? The raising millions out of poverty, increasing lifespan, less death, better education, or more freedom?

"I value not taking someone's stuff, just because I want it."

Then why not directly get what you produce?

"The Marx paper assumes nobody crosses over from being a worker to being a business owner."
Work in a sweatshop or build your own. Also this is impossible for the vast majority