Application of Article 148 of the Civil Code

A, married, cohabits with another woman. are the two governed by co-ownership?

No, there is no co-ownership of A with the other woman unless actual contribution is proved.

If the actual contribution of the party is not proved, there will be no co-ownership and no presumption of equal shares. (citing Agapay vs. CA, et al., 276 SCRA 340).

additional reference

JOAQUINO v. REYES 434 SCRA 260

FACTS: Respondent Reyes is the widow of Rodolfo Reyes. Reyes had illicit relations with petitioner Joaquino. A property in BF Homes Paranaque was executed in favor of Joaquino. Joaquino had no means to pay for this property. The funds used to purchase this property were earnings of Reyes from his position as corporate executive and from a loan secured from
Commonwealth Insurance Corporation. Joaquino and Reyes had 3 illegitimate children.

HELD:
Article 148 is the property regime that will apply in case where the partners have a legal impediment to marry each other. In this property regime, only the
property acquired by them through their actual joint contribution of money, property or industry shall be owned by them in common and in proportion to their respective contributions. The registration of a property in the name of the paramour who had no income whatsoever at the time of the donation by a
husband is tantamount to a donation which is void under Article 87 of the Family Code. The paramour then holds the property under a constructive trust
under Article 1456 in favor of the conjugal partnership of the husband with the legitimate spouse. Status of an illegitimate child who claimed to be an heir to a decedent’s estate could not be adjudicated in an ordinary civil action such as in a case for recovery of property.

^thanks for the additional info