Let's assume that GHG and carbon emissions are a reality. What would the free market solution be? It wouldn't be a carbon tax, because that involved tax. It wouldn't be cap and trade, because that implies government force on the cap.
Would the free market solution be freely trading the emissions if an individual/entity desires? Private organization would be able to limit via a cap and trade system, because it is private and volunatry to join.
This would be handled just as any other pollution would be handled; it would be enforced through property rights. Once you prove how much damage was caused on your property through the carbon pollution emmited from a factory, you would be able to sue them for damages.
Those factories that emit less pollution will have lower costs compared to factories which emit lots of carbon, therefore the cleaner companies would be able to compete with the polluting companies much easier.