I understand your concern, but there is no statutory prohibition against cohabitation. This means that it is a matter for judicial oversight, so each judge can make the decision that they feel is in the child's best interest.
Here is the statute that gives the judge that authority (notice the breadth of the last factor):
(c) Determination of child's best interests. In determining the child's best interests for purposes of allocating significant decision-making responsibilities, the court shall consider all relevant factors, including, without limitation, the following:
(1) the wishes of the child, taking into account the
child's maturity and ability to express reasoned and independent preferences as to decision-making;
(2) the child's adjustment to his or her home,
school, and community;
(3) the mental and physical health of all individuals
(4) the ability of the parents to cooperate to make
decisions, or the level of conflict between the parties that may affect their ability to share decision-making;
(5) the level of each parent's participation in past
significant decision-making with respect to the child;
(6) any prior agreement or course of conduct between
the parents relating to decision-making with respect to the child;
(7) the wishes of the parents;
(8) the child's needs;
(9) the distance between the parents' residences, the
cost and difficulty of transporting the child, each parent's and the child's daily schedules, and the ability of the parents to cooperate in the arrangement;
(10) whether a restriction on decision-making is
appropriate under Section 603.10;
(11) the willingness and ability of each parent to
facilitate and encourage a close and continuing relationship between the other parent and the child;
(12) the physical violence or threat of physical
violence by the child's parent directed against the child;
(13) the occurrence of abuse against the child or
other member of the child's household;
(14) whether one of the parents is a sex offender,
and if so, the exact nature of the offense and what, if any, treatment in which the parent has successfully participated; and
(15) any other factor that the court expressly finds
to be relevant.
So if the court feels that the cohabitation results in an unstable environment, contradiction of how the child is being raised, or some other legitimate concern, the court may forbid the child from spending the night at the residence of the cohabitating party. They are more likely to allow it if it is religious based, or if the family is conservative in all respects as that helps prevent against frivolous requests.
However, absent a court order, there is no legal preclusion re: cohabitation.
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