Erm, what exactly do you mean by "tethers" here? I'm assuming you're not referring to a physical tether, so I'm kinda lost.
OK so very briefly because Fate is dense:
Heaven's Feel/The Third Magic is basically a turbocharged, look ma no Mana limit, Spark from Magic the Gathering combined with the concept of Perfect Reincarnation some writers like to use, think Hao from Shaman King, that the Einzbern family were created to recover by the disciples?/apostles?/followers?/creepy fans? of the woman who had achieved it.
With the creation of the Einzbern family as a Magus family with Justeaze Lizrich von Einzbern, gesundheit, which caused the suicide/abandonment of the project by it's progenitors left the newly minted Magus family in a bit of a conundrum as while Justeaze herself could theoretically master the Third Magic if she put in the work she couldn't share it with other people.
So since she was created to help everyone get it she instead put in centuries of work to figure out how to do that which ended with her stepping into what is basically a giant magical mortar and pestle to be ground into gristle that was then used to create a Red Grail, the so called Greater Grail, that would serve as a basis for a ritual to tap the Root's quirk of the Throne of Heroes, a layer of Souls/a part of the Root yet separate from it that actually has differentiated Souls instead of just vague Soul-stuff in it, to acquire/create Souls that could be used to gain the Third Magic for all of Humanity/Access to the Root.
Now the Greater Grail works by creating a Lesser Grail which is made from a clone of Justeaze wearing a powerful Mystic Code called the Dress of Haven, usually as some people have figured out other means of making these, that draws in Souls of living people, usually Magi but not always, as catalysts for performing a ritual for summoning a mote of Energy directly from the Root that acts as a Mirror Neuron that then copies over the information from the Throne of Heroes and creates a new "Soul" that is then anchored by the Soul of the catalyst.
The Souls on the Throne of Heroes are called Heroic Spirits, while the "Souls" are called Servants. The Souls of catalysts are usually only called Designated Masters when they are called anything at all as what matters is the fact that the Souls Servants are anchored to are called Masters. There are 7 Servants in a "normal" Holy Grail War and without extra Servants a winning Magi has to kill their own Servant to get at the Greater Grail so if a Servant wishes to live again the ritual gets fucked over as their "Soul" becomes their actual Soul.
So why are Holy Grail Wars such tragedies? Because the Greater Grail has gone horrifyingly right when it was built by people who didn't do their homework properly.
First of all the Lesser Grail wasn't supposed to be a thing, but the people using the Greater Grail for the first time while having Justeaze's complete notes realized they were not in fact on the same page on the topic of what to use the Greater Grail for and so decided to fight it out amongst themselves and also bring in reinforcements since they needed Seven "Souls" to activate the Greater Grail. This ended in a wet fart as while the Three Families, The Tohsaka, The Matou and the Einzbern, were willing to fight it out at least one of the four outsider Magi just plain left with their Servant.
This resulted in the Lesser Grail being designed out a clone of Justeaze and a Mystic Code replica of her magical robes the Dress of Haven to serve as a binding of sorts to insure the ritual came to full fruition including adding in the whole wishcraft aspect of it. A murder spree later that got the name The Holy Grail War for the ritual and nothing much of else the Three Families decided to hammer out some hard rules for this ritual that would insure that it actually works.
This meant both designing a selection system for the Masters, designing Vessels for the Servants and insuring that they didn't keep their memories of the War and dealing in the Church as a neutral observer instead of the Einzbern family because they still wanted to participate instead of insuring that everything was working OK.
It's at this point that the Holy Grail War takes on the form known to most people. Fuck ups include:
- Calling/Binding the Knight Classes by Weapons instead of Dog/Lion/Kite of War that they actually are.
- Designing the Archer Class to almost always be at odds with their Master by designing a magical artillery Vessel that can't be counter-fired on easily from a Magus' point of view.
- Designing the Saber Class as the on paper "superior" Class.
- Designing the Berserker Class with the goal of being able to summon mindless beatstick slaves. And spectacularly failing at it in so many ways the moment the summoning formalcraft isn't done by one of the people who have Justeaze's original notes.
- Designing the Assassin class to only work with the 19 Hassan-i-Sabbah to the best of their ability without testing out what that actually means. Cue at least the Hassan of Serenity causing the Assassin Vessel to work on powerful Wraiths and not just Heroic Spirits resulting in an accidental recursion formalcraft in the Throne of Heroes that causes powerful Wraiths, or powerful amalgamations of them like Jack, summoned as Servants to ascend to being actual Heroic Spirits.
- The sheer ivory tower stupid in designing the Caster Class as a Class for Magic users and not once thinking that any type of Magic user would qualify because the only magic use they could imagine was one done by a Magi.
- The sheer WTF of the Rider class which only has the Heroic Spirit spent most of their life with a mount. Any sort of mount. Hot air balloons qualify. Magical chauffeurs for the win I guess.
So that's about it for the text and subtext needed to explain the context/subtext?/metatext of what a tether is:
Put simply the tether is the formalcraft holding the Servant in the living world. It is made of the regret the Heroic Spirit/Wraith has about the living world combined with the piece of legend from the Throne of Heroes copied onto the "Soul" catalyzed by the Master to create a Vessel for a Servant to keep them in the living world and then use them as a reagent stored in the Lesser Grail until a wish is made and 7 Servants are dead to activate the Greater Grail. It is the Servant side of the Servant-Master bond.
The pure Metatextual implications of this is that a tether can be unmade by having a Servant resolve the regret and still count both as the death of the Servant as the formalcraft holding the "Soul" unravels leaving it viable for use in the Holy Grail ritual and as a victory because the Heroic Spirit would, because of a quirk of how Souls work in Fate, get back the memory of the resolution past the now non-existent memory block on the Throne of Heroes and be improved by it.
The only problem with that being that most Servant's wishes are not the resolution of their regrets, but some sort of avoidance of it.
Diarmuid Ua Duibhne has his wish and the resolution of his regret lined up, but he has other issues.
Hope I explained this well enough and I welcome criticism because I am most definitively not a Fate otaku.