Patching and Plastering - Replace the tape or just new mud?

I just purchased a house with a "bonus" room above the garage. One of the tape joints at the ceiling and one at the top of the opposite knee wall have failed along their entire length of 23 feet. Basically one side of the tape has loosened, leaving a visible crack at the edge of the tape. The "loose" side peels back quite easily while the other side seems tight. It appears that there was next to no mud applied to one side of both joints. Also, I noticed that this was a plastic, or perhaps fiberglass, tape I have not seen before. I wonder if it was self adhesive or some special purpose tape.

Do I need to pull the tape completely off and replace it, or can I just peel up the loose side and force some mud under and expect it to stick? If I pull the tape down, how about the old mud?

Further, this will be my first experience with a texture. I assume I will have to scrape off the knockdown back far enough to make a proper tape joint. Should I go further back to help disguise the repair or only as far as necessary? Are there any specific things I need to look at in the materials to get the best match with the existing texture?

While you might get away with forcing mud under the tape to "reglue" it, it would be a lot better to remove the failing tape and retape. It sounds like they used the 'sticky' [mesh] tape. It's prone to failure, paper tape always does a better job.

It wouldn't hurt to scrape the texture down some but it isn't necessary. The angle will hide any minor build up of joint compound. Knockdown texture is made by thinning down j/c and spraying it as a splatter coat thru a hopper gun. As the j/c sets up it's knocked down with a wide knife. The consistency of the mud, orifice used on the gun and air pressure used, all play a part in how the texture will look. It may take a little experimenting to match the existing texture.

Yeah, I figured removal was the "right" way, and I've pulled one down, but I may try the cheat method on the other joint. It'll be easier to reach if I have to fix it again, for good. I just hate to waste the good half of the job...

Anyway, the tape use was not glass mesh which I tried once just to see how it worked. If that is the only fiberglass type out there, then this must be just plastic, thinner than a corner bead and more flexible. I just had not seen tape like this before and wondered at one point whether the smooth surface was not very well suited to adhering to the mud. If it means anything, it had graduation marks, like a measuring tape.

By the way, when two seams meet, is it better to overlap the tapes or butt them together?

As for the texture, it sounds like some practice on cardboard outside might be a good idea before using it inside. I'm a bit of a perfectionist and I would spend a long time fretting about how "obvious" the patch looked.

Does it matter which direction I pull the knife, along or across the seam?

Thanks for your comments.

There is a tape that is designed for 'off' angles, I'm not real familiar with it but that may have been what they used. I'm a painter, not a drywall finisher;)
I usually butt the tape. The knife should be pulled with the direction of the tape.