The legitimate superior appointed by the Vatican, Marie of the Incarnation, emphasised in a statement published on Tuesday that the election of a superior under the supervision of the Pius Brotherhood in August was neither permitted nor validly carried out.
After months of conflict with their diocesan bishop and the Vatican Dicastery for Religious, the sisters had placed themselves under the supervision of the Priestly Society of St Pius X, which is not in full communion with the Catholic Church. Mother Marie is now discussing the next steps with the Holy See.
According to media reports, the nuns have also transferred the assets of their convent to a foundation organised under private law.
According to the TV station CBS News, the "Friends of the Discalced Carmelite Nuns of Arlington" foundation has the task of managing and preserving the assets transferred to it, including the monastery's land and property, for the benefit of the community of sisters.
According to canon law, the assets of religious congregations are church property.
The members of the order are therefore not free to dispose of them as they wish and, in particular, cannot simply transfer them to non-ecclesiastical legal entities such as the foundation that has now been established.
The dispute over the convent and its former superior Mother Teresa Agnes has been smouldering for over a year. Mother Teresa Agnes is said to have admitted a breach of her vow of chastity.
As a result, the local diocesan bishop of Fort Worth investigated the matter and had the Carmel searched.
The dispute came to light in May last year after the sisters went public to protest against the bishop's actions.
The Vatican largely rejected the sisters' complaints, but placed the convent under the authority of Sister Marie of the Incarnation, the president of the relevant Carmelite order, instead of Bishop Olson.
However, the nuns refuse to recognise her authority.