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A star A goes through the body of another star B, or is swallowed by B. There is no tidal disruption.

It sounds like kerzan. Does anybody recognise this pronunciation?

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Were you listening to somebody with a southern drawl say this? – TheBluegrassMathematician Apr 22 '14 at 1:27
He just forgot its pronunciation. You can take a look at Py-ser's post below. – questionhang Apr 22 '14 at 3:05

The term you're searching for is probably accretion (I guess it might sound similar to a kerzan if the speaker had a cold or the listener couldn't hear it well enough for other reasons). This term is not limited to close binary stars (accreting binaries) cannibalizing each other because they orbit each other at such proximity that the L1 Lagrange point between them extends into their own Roche lobe, they form accretion disks connected through an accretion stream, and they feed off each other (thus why this is often referred to as cannibalism):

              enter image description here

   Artist's conception shows an accreting binary at the center of the globular cluster NGC 6624 (Source: U. of Tennessee AstroWiki)

Some examples of accreting binaries are symbiotic novae, X-ray binaries, microquasars, X-ray pulsars, and so on. See links for more examples and explanations. Other examples of accretion include stellar accretion of planetary materials of a planet in a low altitude orbit around its parent star, formation of planets via accretion of materials of the protoplanetary disk, active galactic nucleus (AGN) like quasars (could also sound similar), novae, and e.g. the Earth still accretes on average ~ 110 metric tons of new materials per day, too.

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Curious, what would happen if hypothetically the two stars weren't close but were on a direct collision course with each other? – TheBluegrassMathematician Apr 22 '14 at 1:06
May explode as a supernova. May generate a black hole. May create a bunch of planets. There are many possible outcomes. – Florin Andrei Apr 22 '14 at 1:38

Is it possible you refer to a Thorne–Żytkow object ? This is literally a star within another star, more specifically a compact object (Neutron star) inside a giant star (red giant or supergiant).

enter image description here

As far as I know, this is an only theoretical object, it has never been observed.

In the original paper from the authors, they provide some observable characteristics of these objects, like some peculiar emission lines. You can imagine the chemistry of such an object would be really unique.

Here an article for a possible identification link 1, very recent work (2014).

And here some calculations link2.

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So when this thing goes supernova, what happens to the neutron star? And what about black hole formation? But I must say that is one of the cooler things that I've seen. – TheBluegrassMathematician Apr 22 '14 at 1:26
"What" goes supernova? :) You have two different objects here. One of them is already collapsed. – Py-ser Apr 22 '14 at 1:39
Well definitely the red giant. It's going to collapse in on itself so what does that mean for the inner neutron star? Is it super compressed into a black hole? – TheBluegrassMathematician Apr 22 '14 at 1:52
@Py-ser Yes,TZO is what I heard. I do not know why there is not an accretion disk formed. Because generally there is angular momentum and a disk is necessary. – questionhang Apr 22 '14 at 2:52
@Py-ser do you know any similar paper that study how a small black hole goes through a star? – questionhang Apr 22 '14 at 2:57

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