Then, however, they do not usually have communion between these bishops, despite accepting that they are wholly legitimate lines of succession. The bishops do not share their altars with others [see the correction offered in the comments]. What they usually do is incardinate, that is bring the other in. The bishop shares with other bishops of the same (mini) Church, and priests are incardinated under them. To connect one Church with another is to develop a close relationship, one being the senior partner of the other.
This is why The Liberal Rite does not even share communion with the Liberal Catholic Church International, even though the LCCI was involved in consecrating those bishops who make up The Liberal Rite.
The last paragraph is factually wrong, certainly from The Liberal Rite and Independent Liberal Catholic Fellowship points of view, and it was a Canadian strand of the LCCI that was involved in The Liberal Rite consecration. Liz Stuart later moved into the British and Irish LCCI.
The explanation being given formed part of a wider explanation about the mess of the Anglican Communion, and if it is becoming a Church above Churches, and formal agreements of recognition and intercommunion made with other Churches that of course stay as other Churches.
So there is a need to clarify (and this is checked) as it is important not to misrepresent groups.
First of all, in a number of the liberal groups, and specifically the Independent Liberal Catholic Fellowship, the altar tables are free in terms of offering sacraments.
Secondly, altar tables are shared, but agreements of intercommunion are indeed problematic. It is the setting up of formal agreements that is the problem, as they suggest that without them there is no intercommunion - when there is. The idea and practice within many sections of the Independent Sacramental Movement is to free matters, not add barriers.
This was what I was trying to indicate for the other argument (if in a round-about way). Having a Communion does not involve a tendency to centralisation, as seen with Rowan Williams, which in effect transfers a Church upwards. The Independent Sacramentalist Movement has these small named Churches, situations of recognition, but they remain the masters of their own organising.
Incidentally, and as a qualifier to all of this, the Liberal Catholic Church International does not allow concelebration with other clergy, and it has introduced something of a creed. So what I wrote is correct regarding them, but not regarding others.
My own broader interest in this, I suppose, goes back to the inspiration of J. M. Lloyd Thomas, and James Martineau, with the further Catholic development of the reinvention of a Presbyterianism without the Puritanism, this development drawing from the Unitarian side that has been sidelined by the Unitarians themselves; as well as, on the Anglican side, the potential for the breadth found within the more symbolic approach of a conserving liturgical approach and, within that (and at the same time), a hopefully confident and freer theological expression.
There is also an area of debate here between the non-realist (which does not mean not realist!) and the apophatic (the via negativa), and so the relationship between spirituality and theological content. My view is about having the widest possible theological content consistent with a spirituality that works - one that includes the ethical Jesus, and a creative working of a freedom in a symbolic gift exchange between the spiritual and the material in people as individuals and as a collective.