ROME — After one-and-a-half years of feeling their bond deepen, after coffee meetups and French study sessions, Oleh Kindiy leaned in close to his girlfriend in a mostly quiet chapel and offered her a ring. She said yes. But asking for marriage was just his first question.

Because soon after, Kindiy sat down with her again. He told her, this time, that he was interested in becoming a priest in the Ukrainian Greek Catholic Church. If he started down that path and enrolled at a seminary, he’d see her initially only on weekends. And that was just the first sacrifice they’d have to make together. Was that okay?

“I wanted to be honest with her,” said Kindiy, now 41.

Her agreement helped to launch, for Kindiy, the kind of life impossible in most parts of the Catholic world — that of a priest who is married, not celibate, and who divides attention between his family and his clerical duties.

Catholic leaders have long resisted the idea of married clerics in the mainstream Latin Church, considering celibacy an essential element for devoted priests, and this week Pope Francis declined to approve the ordination of married men in the Amazon region.

But in parts of Eastern Europe and the Middle East, in relatively small and distinctive branches of Catholicism that are loyal to the pope, married clerics are the norm — and their lives represent an alternate version of the Catholic priesthood.

In that version, the typical priest is one like Kindiy: somebody who juggles his duties, sometimes has to rearrange his schedule, and watches movies or plays chess with his kids on days he gets home early enough. His family gathers together with all the others at a church cafe after Mass. He has asked parishioners to serve as godparents. Sometimes, Kindiy takes his 10-year-old son on visits to the sick. If they ask for a blessing, his son serves as cantor.

Beyond the day-to-day logistics of being a priest and a father, there is also a sense among married priests that the broader Catholic fears are unfounded. Married priests say they admire the tradition of celibacy, but they also feel like having a family can help them be more integrated in the community and understand of some of their parishioners’ problems.

“Basically priests and lay people are on the same level,” said Kindiy, who lives in the western Ukrainian city of Lviv and serves at a large parish where eight of the nine priests are married. “Married priests are not above the folks. My wife is friends with the parishioners. My kids are growing up with parishioners’ kids.”

The rules of the Ukrainian Greek Catholic Church might feel contradictory to Roman Catholicism, but they show how Catholic leaders have generally been willing to tolerate self-contained exceptions to the celibacy rule.

The Eastern Rite churches represent just 2 percent of the Catholic faithful. The Vatican has also provided celibacy exceptions for already-married Anglican ministers who convert to Catholicism. In 2014, Francis with little fanfare drew up an exception of his own, allowing for married Eastern Catholic men to be ordained outside their traditional territory — including in the United States.

But what popes have been unwilling to do is grant exceptions that would open any pockets of mainstream Catholicism to married priests.

Latin American bishops had proposed the ordination of married men in the Amazon as a way to address drastic clerical shortages and help remote areas that sometimes go years without Mass. Traditionalists were vehemently opposed, arguing that the exception would set a precedent that could revolutionize the priesthood. In one representative critique, conservative Cardinal Robert Sarah told the National Catholic Register that a married priest is “unable” to be “totally and absolutely given to God and the Church.”

“Right now, we are in a period of retrenchment,” said the Rev. Paul Sullins, a married former Episcopalian priest who converted to Catholicism in 1998. “I don’t think the Latin Rite Catholic Church or the vast majority of dioceses are anywhere near the place of accepting married priests on a regular basis.”

The prospect of married clergy presents additional complications, according to experts who have studied the issue. Married priests tend to cost the church more in terms of housing and health care. If they have young families, it can be harder for them to move between parishes midcareer.

And allowing for marriage may not be enough to attract young people to the priesthood. Archbishop Sviatoslav Shevchuk, the head of the Ukrainian Greek Catholic Church, told reporters last year that his rite, too, is dealing with shortages.

“It’s not like marriage is some automatic cure,” said Adam DeVille, an associate professor at the University of Saint Francis in Fort Wayne, Ind., who has edited and contributed to a forthcoming book on the married Catholic priesthood. He said that the strains of the priesthood can also take a toll on families and marriages.

“I know that from seeing people really, really struggle,” DeVille said.

In Ukraine, a man is allowed to marry before ordination, but not after. And the wife must officially sign off before her husband can enter the priesthood.

Kindiy is not the first Catholic priest in his family. His great-grandfather was one, too — and was deported to Siberia to during the Soviet era. After Ukraine became independent in 1991, churches emerged from the underground, and Kindiy, who had grown up in a secular environment, found himself curious.

“This was my question: Why are we suffering? What is the way out of it? In the 1990s, we didn’t really have much money for food,” Kindiy said. “It was economic difficulty. The answers I heard in church really called to me.”

Kindiy never had to choose between marriage and the priesthood — “married priests are so natural for me,” he said — but in the early 2000s, he studied at the Catholic University of America in Washington, surrounded by aspiring priests who’d be bound to celibacy.

“I respect the Catholic insistence on celibacy,” Kindiy said. “They lived together, joked together. I liked to hang out with them. But I realized, after the seminary, they would have to go home, where they would be alone. That is something that is challenging.”