Hello At Trent,
That link inspired the question

It seems that the US head quarters is in Chicago, yet they are under the Archdioceses of Florence. I sensed some possible associations or possible mis-associations with SSPX.
I looked at some of their links, have been looking for a Tridentine Mass in the midwest, and it seems as though some parishes are “under” the “care” of this institute.
I do not understand the structure here.
Happy Easter, SP
The Institute, often known by their latin acronym ICRSS, for Institutum Christi Regis Summi Sacerdotis, is a papally established order of priests and religious, nominally assigned to the care of the Archbishop of Florence, Italy, since he welcomed their mother house; the society is, however, actually independent of His Eminence, electing a superior general according to their constitution. His Eminence, however, also granted them, as an order operating within his see, their special distinctive choir dress (which serves as a habit).
They were established in 1990, for mostly the same purpose as the SSPX had been: to preserve the pre-vatican II praxis.
All their priests are trained for the 1962 missal, now called the EF or Extraordinary Form, but also all are familiarized with the OF (ordinary form) missal.
They have, however, maintained fidelity to the magisterium and the pope. They are canonical and non-schismatic, and are valid and licit.
A parish staffed by the ICRSS is a diocesan parish of the local Roman Rite diocese. The bishop has asked (and made pay arrangements for) the ICRSS to provide priests (and sometimes also deacons) to staff those parishes for him. The Society then operates that parish per the terms of the contract.
Such assignments are nothing terribly unusual; the Franciscan and Dominican friary orders often accept such contracts, provided community living can be maintained and the particular house’s charism is compatible. Jesuits also take such assignments, tho not always in community. Other orders also do so in varying situations and with certain restrictions. It just isn’t obvious when its not a group with a distinctive vesture and/or habit.
The ICRSS have, like the franciscans and dominicans, adopted a system of multiple different sub-orders: they have their ordained society of priests and deacons, and it’s “novices” are seminarians; they have a sub-society of oblates, who while not priests nor seminarians, assist the society in its duties, and another of nuns who do likewise, and a “third order” (tho not called such) for laity who wish to become affiliated with them. Further, many priests remain diocesan priests of their home diocese, but take training from the Institute, and may even be awarded symbols of that affiliation, without formally being under the order’s incardination.