"she believed in the free union"
The unfree union? (U.S. President's marriage, 1886)
Public DomainThe unfree union? (U.S. President's marriage, 1886) - Credit: T. de Thulstrup

A  free union describes two people living in partnership without public recognition (either civil or religious) of the fact.

Today, we would simply say 'living together', although during the period in which Tipping the Velvet is set, it would probably have been described as 'living in sin'.