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Where does fusion occur then? At the surface of the neutron star? Or is the star supported by the kinetic energy of matter falling onto the inner star?
Both. Red giants exhausted supply of hydrogen in the core, so fusion is switched to shell. This causes pressure/temperature of red giant to drop dramatically and radius to balloon. Neutron stars are incredibly dense, so under certain conditions collision of red giant with neutron start could work out.
Could use them as cardinal points in a galactic map, if they're that rare. Maybe they are actually galactic navigational beacons!
I wished the article went into more detail about how it could form. My hazy memory of astronomy from college, I thought matter falling onto a neutron star from a companion star was how a type IIa supernova occurred. How would the neutron star be "swallowed" without triggering the nova?
original paper: http://adsabs.harvard.edu/full/1989ApJ...346..277E discussing the concept. Fusion would happen on the surface of the neutron star. An interesting concept. I suppose if a neutron star entered a gas cloud you could get one of these.

Edit: Not the original paper but a follow on how these stars would work, refuting other papers that said they were impossible :-)

Found more detailed explanation of what they are: http://cow.physics.wisc.edu/~ogelman/guide/tz/
Great link!

The TL;DR:

Because the density of the neutron star and the Red Giant are so different (average Red Giant density is about the same density as water), the neutron star can keep a distinct structure/orbit for 1000 years or so. When it gets to the core, fusion occurs in a halo around the neutron star instead of through the normal sort of fusion you'd get in a Red Giant core. This leads to a different ratio of nuclear isotopes, which is how you can determine these objects aren't typical Red Giants.

The resulting objects survive around 70 million years before the neutron star core absorbs enough mass to turn into a black hole (one of the other linked papers in this thread had some discussion on how long they might be stable). However, the predicted rate of birth/death of these objects indicates there may be a few of these in the galaxy at any given time.

So as a follow up question. What happens if a black hole falls into a super nova? And bonus question: what happens if a black hole falls into a neutron star?
Thinking as a physics generalist but not an astronomer, I'd describe these things as:

1. Super nova happens near a black hole. Stuff (including photons) that emanate with a speed and direction sufficient to overcome the escape velocity of the system will have happy sailing. Photons at the speed of light, massive stuff slowed down. The rest will be sucked into the hole. The result would be a directional burst that favors photons over massive material.

2. Same as saying that a neutron star falls into black hole.