BJC:
Sure, you can make synthetic hydrocarbons by the Fischer-Tropsch reaction from syngas produced from coal. Or syngas from natural gas, wood, corn stover, coconut husks or just about any other source of carbon EXCEPT carbon dioxide or carbonate rocks. The latter are mere sources of carbon, NOT FUELS! F-T technology has existed for the past eighty years or so. Chemists and chemical engineers are really good at reconfiguring atoms into the molecules we want, IF you have enough energy to ADD to drive the required processes and separate the products. What's your point?
Going the syngas/Fischer-Tropsch route, regardless of the source of carbon, is (with a very few exceptions) energetically enormously wasteful relative to taking those same hydrocarbons directly out of crude oil or natural gas/gas liquids by simple distillation and/or relatively simpler chemical transformations to make chemical feedstocks. "Energetically wasteful" here also means "generating far more moles of CO2 per kg of plastic produced".
If the intent is to use the synthetic products eventually as fuels, going the syngas/F-T route is energetically idiotic relative to simply burning the natural gas, coal, corn stover etc. itself as a fuel for, say, stationary applications like heating and electrical generation, reserving the existing crude resources to generate transporation fuels and chemical feedstocks.
The only exception where F-T makes sense is perhaps as a use for "stranded gas"- natural gas discovered and produced too far away from major markets to use it within economical reach of a pipeline, such that wasting half of it to make liquid products for burning later makes economic sense. It also makes energetic sense relative to simply flaring this gas.
Long before we need to make massive use of F-T technology to make our liquid fuel needs from coal, we'll be making heavier and heavier investments in recovering heavy oils from tarsands and oil shale resources. Converting these to liquid fuels and chemical feedstocks is far less energetically wasteful than going the coal/syngas/F-T route, and we have HUGE reserves of these resources.
The only question is: can the planet survive us burning all these fossil resources to fuel the expansion of the western world's idiotic, energy-addicted lifestyle to the developing world?