Archbishop Lefebvre explaining that his bishops are not claiming ordinary jurisdiction:

We are striving to act in such a way that we cannot be reproached with the bishops' being given a territorial jurisdiction, in such a way that there is no bishop being attributed to such and such a territory. Of course, it's only normal that a French bishop should go to France, and that a German-speaking bishop should go to Germany, but from time to time, we try to bring about an exchange in order to head off that accusation. Of course, it is normal that in the United States, Bishop Williamson should give the confirmations. But Bishop Fellay went to give confirmations in St. Mary's, Kansas, and so one cannot say that the United States are the domain of Bishop Williamson. Bishop Fellay also went to South Africa which had previously been visited by Bishop Williamson. As for Bishop Tissier de Mallerais, he went to South America and to Zaitzkofen in Germany. So, we are striving to establish this principle, that there is no territorial jurisdiction.

From Fideliter:

Bishop Tissier de Mallerais: Firstly, I was assured that, by such a consecration, even carried out against the will of the pope, neither Archbishop Lefebvre nor myself nor my confreres were creating a schism, since the Archbishop did not intend to assign us any jurisdiction, or a particular flock. "The mere fact of consecrating a bishop [against the will of the pope] is not in itself a schismatic act," declared Cardinal Castillo Lara (President of the Pontifical Commission for the Authentic Interpretation of Legislative Texts; quoted from an interview given to the newspaper La Repubblica, 10 July 1988.) a few days after the event; and Fr. Patrick Valdrini also explained, "It is not the consecration of a bishop [against the pope's will] that creates a schism...; what consummates the schism is to confer upon that bishop an apostolic mission." (Doyen of the Faculty of Canon Law of the Catholic Institute of Paris; interview appearing in Valeurs Actuelles, 4 July 1988.)

Fideliter: But didn't Archbishop Lefebvre confer upon you an apostolic mission?

Bishop Tissier de Mallerais: Archbishop Lefebvre told us: "You are bishops for the Church, for the Society; you will give the sacrament of Confirmation and confer Holy Orders; you will preach the faith." That is all. He did not say, "I confer these powers to you"; he simply indicated to us what our role would be. The jurisdiction that he did not give us - which he could not give us - and which the pope refused to give us, has been supplied by the Church, who gives it to us because of the state of necessity of the faithful. It is a suppletory jurisdiction, of the same nature as that which is accorded to priests by Canon Law in other cases of necessity. An example would be the jurisdiction to administer the sacrament of confession validly in the case of common error or positive and probable doubt, of right or of fact, about the jurisdiction of a priest (canon 209). In such a case, the Church has the habit of supplying the jurisdiction that might be lacking to the minister: "Ecclesia supplet."

Fideliter: So, by receiving the episcopal consecration in such circuмstances and by exercising its power, you were able to be sure that you were not usurping any jurisdiction.

Bishop Tissier de Mallerais: Yes, no ordinary jurisdiction. Our jurisdiction is extraordinary and suppletory. It is not exercised over a determined territory, but case by case over the persons who are in need: confirmands, seminarians of the Society or candidates to the priesthood recommended by other traditional works.

Fideliter: Your consecration, then, Your Excellency, did not create a schism.

Bishop Tissier de Mallerais: No, not in any way. But a touchier question was talked about as far back as 1983, when Archbishop Lefebvre, confronted with the 1983 Code of Canon Law published by John Paul II, began to seriously consider consecrating one or more bishops: would these bishops, not recognized by the pope, be legitimate? Would they enjoy the "formal apostolic succession"? In a word, would they be Catholic bishops?

Fideliter: And that is a more difficult question to resolve than the one about jurisdiction, you say?

Bishop Tissier de Mallerais: Yes, because it has to do with the divine constitution of the Church, as all Tradition teaches: there can be no legitimate bishop without the pope, without at least the implicit consent of the pope, by divine right head of the episcopal body. The answer is less evident; in fact, it is not at all evident...unless you were to suppose...

Fideliter: Your Excellency, certainly you are not a sedevacantist?

Bishop Tissier de Mallerais: No, in fact. But it must be recognized that if we could affirm that, for reasons of heresy, schism, or some defect in the secret election, the pope was not really pope, if we could pronounce such a judgment, the answer to the delicate question of our legitimacy would be clear. The trouble, if I can so express it, is that neither Archbishop Lefebvre nor myself were or are sedevacantists.