Final answer:
The correct pronoun for a castle owned jointly by a lord and lady is 'their' as it reflects shared ownership and is gender-neutral.
Step-by-step explanation:
The correct pronoun to use when referring to a castle owned jointly by a lord and lady is their. This is because the possessive case pronoun their indicates ownership by multiple people, regardless of gender. The sentence would read as follows: Either the lord or lady will take you on a tour of their castle. In English, when ownership is shared and includes both male and female owners, the plural possessive pronoun their is appropriate and avoids gender bias.