An archbishop uses a green galero with twenty tassels, 1-2-3-4 on each side (first image below).

I have also read that the archbishop of a primal see (a primate) is entitled to thirty tassels, 1-2-3-4-5 on each side, and that he may interweave gold threads into the cords of his galero (middle image below).

And also that an archbishop who is a Latin-rite patriarch (such as Jerusalem, Lisbon, Venice) is also entitled to thirty tassels, 1-2-3-4-5 on each side, and that he may interweave gold threads into both his galero cords and his tassels (last image below).

http://upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Archbishop_CoA_PioM.svg/200px-Archbishop_CoA_PioM.svg.png http://upload.wikimedia.org/wikipedia/commons/thumb/f/ff/PrimateNonCardinal_PioM.svg/200px-PrimateNonCardinal_PioM.svg.png http://upload.wikimedia.org/wikipedia/commons/thumb/f/fc/PatriarchNonCardinal_PioM.svg/200px-PatriarchNonCardinal_PioM.svg.png

There are copious examples of the first usage, but are the second and third examples borne out in practice, or are they more fanciful heraldic hearsay?

The gold interlacing thing is made up. The 1832 norms put out by… I forget the congregation’s name… specify only green.

Archbishops display a green galero with 20 fiocchi—- you have that correct.

Primates, I’m pretty sure have no special ornaments, but I’d double check with Fr. Guy.

Patriarchs display a green galero with 30 fiocchi—- basically what you have, but without gold

You don’t encounter these much since there are only four of those: Venice, Lisbon, Jerusalem, and the East Indies. Of those Lisbon and Venice always get cardinals… though you will see the green galero while they are "in-waiting" so to speak as the current patriarch of Venice (See here). Here are the arms of the patriarch of Jerusalem.

The Jeusalem patriarch’s arms, at least as rendered by Marco Foppoli, are gorgeous!  I assume (subject to correction of course) that these are the patriarch’s personal arms with a chief for the see?

Michael F. McCartney;97732 wrote:

The Jeusalem patriarch’s arms, at least as rendered by Marco Foppoli, are gorgeous!

I agree.  I much prefer this treatment of "Mary at the foot of the cross" to the capital letter M used in Pope John Paul II’s arms.

Michael F. McCartney;97732 wrote:

I assume (subject to correction of course) that these are the patriarch’s personal arms with a chief for the see?

The chief is an indication that he is the Grand Chancellor of the Equestrian Order of the Holy Sepulchre.

steven harris;97707 wrote:

There are copious examples of the first usage, but are the second and third examples borne out in practice, or are they more fanciful heraldic hearsay?

They are hearsay.

Primates have no special heraldic privileges. Patriarchs use a green galero with cords and 30 tassels all of green.

The confusion about the skein of gold interwoven with green refers to the cording for the tassels they may wear on their saturno or ecclesiastical hat (as opposed to the pontifical hat or galero). In other words, it was worn on an actual hat but not included in their heraldic privileges.

Fr Guy—thanks for the correction.  Are there institutional arms for the patriarchates, similar to a diocese, in addition to whatever personal arms the patriarch may bear?  Or do they follow the papal model?

They all have some corporate coat of arms associated with the diocese that is also their patriarchate. Some choose to marshal these arms with their own some do not.