I have a bill of sale, import papers, registration papers in our name and what most people use here is called a "Carnet de Vaccin" which is just a historial vaccination card. (Annual vaccines are mandatory) Evidently if you are in possession of that and the horse it is good.
Just a note on this thread, we don't have much horse theft here but in my barn we still take precautions. For instance, we don't decorate our boxes nor do we post information that would indicate a horses value or lineage. The only information on the box door is horses nickname, owners name and contact info, emergency info, medical info, farrier and vet info and feeding/turnout instructions. I also keep a file in the house on each horse that is handy which includes owners info and detailed photos. In case a horse were to ever go missing it is readily available to access and distribute the information. I also photograph and note brands and serial numbers of saddles and expensive tack too.
Luckily, knock on wood, we have never had a problem.