This question is always difficult to answer (BTDT with the daylilies); your taste changes and what your "desert island keepers" (or "fire is coming and must save keepers") are will change from year to year. Moreover, commercial availability has to factor into the list. As with the daylilies, some deserving iris cultivars are simply difficult if not now impossible to find from commercial sources.
I note that you have also left out the non-bearded irises (or maybe you didn't?).
But we if are talking strictly about must-have beauty and form in the bearded
irises, then right now, for certain (of those I have seen bloom), I must have:
(large flouncy blooms, always arresting, can hold its own with THAT'S ALL FOLKS; DYKES winner)
THAT'S ALL FOLKS
(I'm still not sure how I feel about most yellow irises, but this one demands
your attention; DYKES winner)
(this should maybe go on the rebloomer list (see below), but it has yet to rebloom for me, so it goes on this list; beautifully formed flowers, fantastic surface texture, PBF; Award of Merit)
(an older cultivar so the blooms are not as large or wide or ruffled as those of more modern irises, but the flowers are still beautiful to me and I love the color contrast of the beard; Award of Merit)
Now I am going to stand up on behalf of the rebloomers. The way you worded this question leaves out most rebloomers, because (from my limited experience and perspective) those generally seem to be lagging behind the spring-only bloomers wrt beauty and form. YET (protesting on their behalf) they certainly deserve a spot in the garden, and at least in this
garden, some would be replaced if lost. So, here are those rebloomers which I have (and which I have thus far BOTH
personally seen AND
experienced rebloom on (I make that distinction because many of my registered rebloomers have not) which go on my Top 10 Rebloomers list:
(most of the yellow rebloomers I have seem to have muddy color, but this one has fairly good color; its rebloom time is all over the map, though; somehow no awards
PURE AS GOLD
(I know this one goes on the list (either list) because I have already replaced it twice (my fault, not the plant's); eye zorching color and decent enough form; Award of Merit)
(As with the daylilies, I generally prefer my irises to be taller and have bigger blooms. This IB deserves to be on the rebloomer list, though, for the multiple rebloom cycles it has. Somehow no awards
, which I think must be an oversight.
(I confess I also like this one for its name, which is why it got invited into the garden despite its smaller size. It brought to mind a doggerel on an old coffee cup which I had bought for its verse, and since lost. I thought I had kept a copy of the verse but couldn't find it, but I did find the images below online on Etsy (and yes, I just re-purchased that vintage cup, for the verse!). Chalk this iris up as another one in homage to our dogs.)
I have many more irises here that I like (and that I would probably try to replace if I lost them), including a great many rebloomers. I am, however, reserving judgment at this time for my remaining slots on either list.