Clearly, the only way Jabba's "somehow" explanation works is by appealing to magic.
Let's see how this "somehow" works.
Doctor Who lent the TARDIS to some future textile experts who first went to the 1st century and produced a herringbone weave piece of fabric, (no other 1st century cloth has been found using a herringbone weave, so they couldn't use any old piece of linen which they found in the Golgotha market) which they ensured was used as the burial cloth of a man who was very oddly proportioned with a head that came to an angle at the top, and who resembled mediaeval paintings of Jesus but who looked nothing like any of the itinerant preachers named Yeshua who were preaching their apocalyptic messages at that time. To be on the safe side, they crucified this poor chap and made his blood flow against gravity even after he was dead. Using some miraculous process unknown up to the 21st century, they imprint an image of the not-Yeshua onto the cloth, making it appear to be painted with mediaeval paints, and making sure it's in an anatomically impossible position.
They had to develop and learn a process whereby they could apply a patch of material from a completely different time to this cloth without disturbing either the herringbone weave or the invisible linen banding. Since this process hasn't been known at any time up to and including the 21st century, perhaps they popped forward in time to learn and develop this?
Then they go to the ten year-long discussion about the sampling area, and hear that it's going to be done along one unobtrusive edge.
Having hidden the cloth away between the 1st century when they manufactured it and the 14th century where it is first reported, they now pop back to the 14th century where they weave another piece of cloth, and apply it as a patch onto the shroud cloth just where the samples will be taken, at one unobtrusive edge, ensuring that this patch is invisible not just to the naked eye but also invisible using every late 20th century technique, and ensuring that the patch matches the herringbone weave and invisible linen banding on the original cloth so finely that it is simply impossible to detect it.
Then they need to fake the d'Arcis memo and all the associated evidence of the mediaeval forgery, and can breathe a sigh of relief and head back to the future, leaving the cloth in the custody of the Church.
Where now? Obviously they need to ensure that the actual, obvious patches on the shroud are always nowhere near their Amazing Invisible Patch™ (hereinafter AIP™), and similarly protect that area from the scorching during the fire, and from the molten silver. That probably requires a few more TARDIS trips. Now, this does beg the question of why these magical time travelling textile experts didn't protect the shroud from the fire or the Poor Clares' handiwork, but I digress.
Finally, they let the radiocarbon dating labs do their thing, and are positively delighted to find that their nefarious scheme to bamboozle the faithful of the world has succeeded. Again, this begs the question of motive; if they could time travel to the 1st century, why not just bring back the real burial cloth of the real Jesus if he existed, complete with an authentication sticker? So much more interesting. And if he didn't exist, then they had no need to do all this fakery anyway. If these time travellers had found Jesus going around and doing miracles but had been unethical enough to want to deny this fact, they could just have altered history while they were back in the 1st century, perhaps by destroying any manuscript copies of the NT stories and later on, sending a message to the council of Nicea telling them that the meetings were all cancelled.
So now every rational person believes the shroud to be a mediaeval forgery, whereas really it's a 1st century/14th century/long time in the future forgery! But a few people have seen through them, and they set up STuRP to try to show that they can't be fooled by such evidence as "radiocarbon dating".
And that's what Jabba must mean when he says that "somehow" the AIP™ rendered the radiocarbon dating unreliable.
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