I've been thinking about this and I don't understand what they are claiming here.

Strange phenomena happen at low temperatures in fluids, and most of them can be traced to a pairing potential that comes into play between the objects in the fluid. These pairs are necessarily "Bosonic", meaning a lot of them can exist in the same quantum state, and the more that exist in that state, the more that tend to pile in, in a self-reinforcing phase transition called a "Bose-Einstein condenstation". One ends up with a "macroscopic Bosonic quantum state".

But "Fermionic" pairs are by definition impossible unless one has integral spin objects interacting with half-integral spin objects. Also, by the nature of Fermions, only one can exist in any given state so one can never have a "macroscopic Fermionic quantum state", even if the indicated pairing were possible.

There are neutron stars, which are made of degenerate Fermionic matter. So "Fermionic condensates" are certainly known.

I think what they must be talking about here is - in a Bose condensate near the phase transition point, there is a "borderland" of uncondensed Fermions (not Cooper paired) that run around in the condensate. Essentially this relatively small amount of free, unpaired Fermions would act like a gas, with the reference vacuum being the Bose condensate itself. Any colder and all the Fermions condense and vanish into this "vacuum", any warmer and all the pairs break up and one has normal matter. So this Fermi gas is happening right at and around the transition point. Nice.

This is very interesting experimentally, but hardly a "new state of matter", since the principles are well understood.