What is the most sexually liberal church denomination? Just a random question because of the conflict between liberal and less liberal churches world wide.
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The Family International, also known as Children of God, is possibly the most sexually liberal religious denomination on the planet.
Statement of Faith: The Family International is an international Christian community committed to sharing the message of God’s love with people around the globe. We believe that everyone can have a personal relationship with God through Jesus Christ, which affords happiness and peace of mind, as well as the motivation to help others and to share the good news of His love. Our fundamental beliefs are generally in accordance with those held by Christians the world over; we also embrace some untraditional doctrines. Our application of the foundation principle of God’s Law of Love that Jesus taught—to love God, and to love our neighbor as ourselves, which He said fulfills “all the law and the prophets” (Matthew 22:37–40)—is a defining feature of our lives and our faith. - Statement of Faith
Although it claims to be Christian, it is branded as a cult:
The Family International (TFI) is a cult which was founded in Huntington Beach, California, US in 1968. It was originally named Teens for Christ and it later gained notoriety as The Children of God (COG). It was later renamed and reorganized as The Family of Love, which was eventually shortened to The Family. It is currently named The Family International.
TFI initially spread a message of salvation, apocalypticism, spiritual "revolution and happiness" and distrust of the outside world, which the members called The System. In 1976, it began a method of evangelism called Flirty Fishing, that used sex to "show God's love and mercy" and win converts, resulting in controversy. TFI's founder and prophetic leader, David Berg (who was first called "Moses David" in the Texas press), gave himself the titles of "King", "The Last Endtime Prophet", "Moses", and "David".
“Loving Jesus” is a term TFI members use to describe their intimate, sexual relationship with Jesus. TFI describes its "Loving Jesus" teaching as a radical form of bridal theology. They believe the church of followers is Christ's bride, called to love and serve him with wifely fervor. But they take bridal theology further, encouraging members to imagine Jesus is joining them during sexual intercourse and masturbation. Male members are cautioned to visualize themselves as women, in order to avoid a homosexual relationship with Jesus. Many TFI publications, and spirit messages claimed to be from Jesus himself, elaborate this intimate, sexual relation they believe Jesus desires and needs. TFI imagines itself as his special "bride" in graphic poetry, guided visualizations, artwork, and songs. Some TFI literature is not brought into conservative countries for fear it may be classified at customs as pornography. The literature outlining this view of Jesus and his desire for a sexual relationship with believers was edited for younger teens, then further edited for children. - The Family International
Here is a recent article about a woman who was born into the Children of God cult:
Children of God BBC News Article 27 June 2018: The Children of God began in the United States in the late 1960s. Its founder, David Berg, told members that God was love and love was sex, so there should be no limits, regardless of age or relationship. "It actively encouraged sexual activities among minors as young as two or three years old," Verity says. Both Verity's parents were active members when she was born. -Children of God cult was 'hell on earth'
With regard to mainstream Christian denominations, there is a growing tendency to condone same-sex relationships and embrace the LGBTQA agenda in an attempt to be “inclusive” and to attract new members. The down-side to this liberal approach is that thousands of ordained ministers and lay-persons are voting with their feet and turning to more conservative Christian churches. The Church of Scotland illustrates this situation. These days it might be easier to ask which Churches still cling to “old-fashioned” biblical standards of morality and sexuality.