There really is nothing "green" about this technology. Natural gas is partially burned to produce CO and H2. A catalyst is then used to form hydrocarbons from the CO and H2. Another catalyst is used to transform the hydrocarbons into ones that do not freeze at low temperatures (bad for airplanes!). Once burned in the jet engine, the hydrocarbons form CO2 and H2O, just like what would have been formed had the natural gas been burned directly.
One can do the same using coal instead of natural gas. This was practiced by the Nazis during WWII and South Africa during the Apartheid years (and still is being used as far as I know).