(CNN) Two Catholic high schools in Indianapolis were facing the same instruction from an archbishop: Fire a teacher who is in a same-sex marriage, or lose the archdiocese's recognition as a Catholic institution.

One refused last week , and is keeping its gay, married teacher. But officials at the other have announced they are firing theirs.

Adding to the tense situation: the two teachers have been married to each other since 2017, multiple sources have told CNN. Their marriage, which the couple shared on social media, led a local Catholic to complain to the archdiocese, the sources said.

The schools different approaches to the teachers appear to be influenced partly on their differing degrees of autonomy and reliance on recognition by the Archdiocese of Indianapolis.

Last week, Brebeuf Jesuit Preparatory School, about 11 miles from Cathedral, refused to fire its teacher in question -- and will no longer be recognized or identified as a Catholic institution by the archdiocese . The school said it plans to appeal the decision to the archdiocese and the Vatican, if necessary.

In a statement Monday, the Indianapolis Archdiocese said all Catholic school employees are ministers of the faith and must abide by all church teaching, including about marriage.

"This issue is not about sexual orientation; rather, it is about our expectation that all personnel inside a Catholic school -- who are ministers of the faith -- abide by all Church teachings, including the nature of marriage," the archdiocese statement reads.

"If and when a minister of the faith is publicly not doing so, the Church calls us to help the individual strive to live a life in accordance with Catholic teaching."

"It is very difficult to part ways" with someone who continues to rebuff the teaching, "but we readily honor the person's dignity and decision," the statement says.

The Catholic catechism teaches that "homosexual acts are intrinsically disordered" but also that LGBT people "must be accepted with respect, compassion, and sensitivity. Every sign of unjust discrimination in their regard should be avoided."

Why Cathedral says it fired teacher

An aerial view of Cathedral High School in Indianapolis.

But to keep its existence as a Catholic school, it must follow the archbishop's guidance, board Chairman Matt Cohoat and President Rob Bridges said.

The school officials laid out what would happen if the archdiocese no longer recognized it as a Catholic institution:

• Cathedral could no longer refer to itself as a Catholic school.

• Cathedral "would lose the ability to celebrate the Sacraments as we have in the past 100 years with our students and community," a celebration "essential to Cathedral."

• Its diocesan priests could no longer serve on its board of directors.

• It would lose its affiliation with the Brothers of Holy Cross.

• Crucially, it would lose its status as an independent nonprofit organization, and therefore could not operate as a nonprofit school.

"In today's climate we know that being Catholic can be challenging and we hope that this action does not dishearten you, and, most especially, dishearten Cathedral's young people," the school's statement reads. "We know that some individuals do not agree with every teaching of the Catholic Church and so their conscience struggles between the teaching and what they believe is right. ...

"Please know that we offer our prayers and love to this teacher, our students and faculty, our Archbishop, and all associated with Cathedral as we continue to educate our students in the Catholic Holy Cross tradition."

Crucial differences between the schools

Brebeuf, which kept its teacher, is in a different situation. It is not merely affiliated with a religious order. It is sponsored and governed by one -- the Jesuits -- and therefore enjoys a degree of independence from the church hierarchy.

Whereas Cathedral depends on the archbishop's recognition to retain its nonprofit status, Brebeuf does not, Kurt Nelson, director of Catholic school programs at Marian University, told CNN affiliate WRTV

"Cathedral school ... isn't part of another network, so if they don't keep their Catholic identity through the archbishop, they're not a Catholic organization, and therefore they would lose their tax-exempt status," Nelson told the Indianapolis TV station.

Cathedral, in its own statement, also broached the difference.

"Because Brebeuf is a specific ministry of the Jesuits, their canonical and nonprofit status is different than ours. Therefore, the two schools cannot function the same way if Cathedral were to receive a similar decree as Brebeuf."

Why Brebeuf says it kept teacher

In a statement , leaders from Brebeuf Jesuit Prep School said they have "respectfully declined the Archdiocese's insistence and directive that we dismiss a highly capable and qualified teacher due to the teacher being a spouse within a civilly recognized same-sex marriage."

The Rev. Brian Paulson, who heads the Midwest Province of Jesuits, said that Brebeuf's instructor does not teach religion and "is a longtime valued employee of the school."

In an interview on Tuesday, Paulson called the archbishop's decision to revoke Brebeuf's Catholic status, an "imprudent and unnecessarily narrow interpretation of enforcing oversight of faith and morals in Catholic education."

"The archbishop would argue that he's doing it out of defense of the faith and preventing scandal," Paulson said, "and I would come and say that I really think that I'm the one preventing scandal in the church."

Paulson said Brebeuf began seeking its own tax exemptions in May, when it became clear that an agreement wold not be reached between the school and the archdiocese.

The school's president and trustees, meanwhile, have called the archdiocese's "direct insertion" into personnel decisions made by a school run by a religious order "unprecedented."

"This is a unique action among the more than 80 Jesuit secondary/pre-secondary schools which operate in dioceses throughout North America."

Leaders of Brebeuf, founded in 1962 as an independent Catholic school, said they have always had control over their personnel decisions.

"Whereas the Archdiocese of Indianapolis may choose to no longer attend or participate in the school's Masses and formal functions, Brebeuf Jesuit is, and will always be, a Catholic Jesuit school," the school's leaders said.

"The Archdiocese has assured us that Jesuit priests may continue to serve at Brebeuf Jesuit and will retain their ability to celebrate the sacraments of the Catholic Church."