Objection 12): How could the entire Church and all the cardinals recognize an antipope, such as in the case of John XXIII (1958-1963)?

Answer: Pope Paul IV declared that Catholics could not accept such a heretical claimant, even if obedience were given to him by "all" – indicating by such a statement that all giving obedience to such an antipope is a possibility. 

Pope Paul IV, Bull Cum ex Apostolatus Officio, Feb. 15, 1559: “6. In addition, [by this Our Constitution, which is to remain valid in perpetuity We enact, determine, decree and define:] that if ever at any time it shall appear that… the Roman Pontiff, prior to his promotion or his elevation as Cardinal or Roman Pontiff, has deviated from the Catholic Faith or fallen into some heresy… (ii) it shall not be possible for it to acquire validity (nor for it to be said that it has thus acquired validity) through the acceptance of the office, of consecration, of subsequent authority, nor through possession of administration, nor through the putative enthronement of a Roman Pontiff, or Veneration, or obedience accorded to such by all, nor through the lapse of any period of time in the foregoing situation;…”

But we’ve already had a situation where all of the cardinals recognized an antipope!  As covered earlier in the book, during the Great Western Schism 15 of the 16 cardinals who had elected Pope Urban VI withdrew from his obedience on the grounds that the unruly Roman mob had made the election uncanonical.  The one cardinal who did not repudiate Pope Urban VI was Cardinal Tebaldeschi, but he died shortly thereafter, on Sept. 7 – leaving a situation where not one of the cardinals of the Catholic Church recognized the true pope, Urban VI.  All of the living cardinals then regarded his election as invalid.[1]

In the 12^th century, Antipope Anacletus II – who reigned eight years in Rome while rivaling the true Pope, Innocent II – gained the majority of the cardinals, the Bishop of Porto, the Dean of the Sacred College, and the entire populace of Rome as his supporters.[2]

See other Answers to the Most Common Objections Against Sedevacantism