Under Caribbean common law, pets are unfortunately treated as personal property (chattels) rather than family members with custody rights. Usually, whoever paid for the pet or whose name is on the veterinary and license records has the strongest legal claim.
That is correct, but in mediation, most attorneys will encourage you to make a practical arrangement outside of court. It is much better to agree between yourselves on care and veterinary expenses than to let a judge treat your dog like a piece of furniture.
