There is variation from state to state, and this is overwhelmingly a matter of state law (or the equivalent in places outside a U.S. state but within the U.S. like Puerto Rico, the District of Columbia, the U.S. Virgin Islands, Guam, American Samoa, the Northern Mariana Islands).
Some states ban cohabitation purporting to be as spouses, other states merely ban applying for a marriage license when you are already legally married.
For example, Colorado's bigamy laws are as follows:
Colorado's Revised Statutes provide at § 18-6-201, entitled "Bigamy" that:
(1) Any married person who, while still married, marries or cohabits
in this state with another commits bigamy, unless as an affirmative
defense it appears that at the time of the cohabitation or subsequent
marriage:
(a)The accused reasonably believed the prior spouse to be dead; or
(b) The prior spouse had been continually absent for a period of five
years during which time the accused did not know the prior spouse to
be alive; or
(c) The accused reasonably believed that he was legally eligible to
remarry.
(2) Bigamy is a class 6 felony.
Then, it provides in § 18-6-202 entitled "Marrying a bigamist" that:
Any unmarried person who knowingly marries or cohabits with another in
this state under circumstances known to him which would render the
other person guilty of bigamy under the laws of this state commits
marrying a bigamist, which is a class 2 misdemeanor.
And, finally, it provides in § 18-6-203, entitled "Definitions" that:
As used in sections 18-6-201 and 18-6-202, "cohabitation" means to
live together under the representation of being married.
This statute would probably criminalize the relationship contemplated in the question.
But, not all state bigamy laws are so broad.
Utah's bigamy statute is narrower. It states
Effective 5/4/2022
76-7-101. Bigamy -- Penalty -- Defense.
(1) An individual is guilty of bigamy if:
(a) the individual purports to marry another individual; and
(b) knows or reasonably should know that one or both of the
individuals described in Subsection (1)(a) are legally married to
another individual.
(2) An individual who violates Subsection (1) is guilty of an
infraction.
(3) An individual is guilty of a third degree felony if the individual
induces bigamy:
(a) under fraudulent or false pretenses; or
(b) by threat or coercion.
(4) An individual is guilty of a second degree felony if the
individual:
(a) cohabitates with another individual with whom the individual is
engaged in bigamy as described in Subsection (1); and
(b) in furtherance of the conduct described in Subsection (4)(a),
commits a felony offense, or for Subsection (4)(b)(xiii), a
misdemeanor offense, in violation of one or more of the following:
(i) Section 76-5-109, child abuse;
(ii) Section 76-5-109.2, aggravated child abuse;
(iii) Section 76-5-109.3, child abandonment;
(iv) Section 76-5-111, abuse of a vulnerable adult;
(v) Section 76-5-111.2, aggravated abuse of a vulnerable adult;
(vi) Section 76-5-111.3, personal dignity exploitation of a vulnerable
adult;
(vii) Section 76-5-111.4, financial exploitation of a vulnerable
adult;
(viii) Chapter 5, Part 2, Criminal Homicide;
(ix) Section 76-5-208, child abuse homicide;
(x) Chapter 5, Part 3, Kidnapping, Trafficking, and Smuggling;
(xi) Chapter 5, Part 4, Sexual Offenses;
(xii) Section 76-7-201, criminal nonsupport;
(xiii) Section 76-9-702.1, sexual battery;
(xiv) Title 77, Chapter 36, Cohabitant Abuse Procedures Act; or
(xv) Title 78B, Chapter 7, Part 8, Criminal Protective Orders.
(5) It is a defense to prosecution under Subsection (2) that:
(a) the individual ceased the practice of bigamy as described in
Subsection (1) under reasonable fear of coercion or bodily harm;
(b) the individual entered the practice of bigamy, as described in
Subsection
(1), as a minor and ceased the practice of bigamy at any time after
the individual entered the practice of bigamy; or
(c) law enforcement discovers that the individual practices bigamy, as
described in Subsection (1), as a result of the individual's efforts
to protect the safety and welfare of another individual.
This statute would probably not apply to the situation in the question.