Being embarrassed by what one's spouse says is sometimes a subtle undertone of ownership. But a person doesn't own their spouse.
Why would you be embarrassed by what someone else says, as if you're responsible for their actions? A spouse isn't a child, they are a whole entire separate person.
It's like when I, a female and a wife, say something that the men don't agree with - they look at my husband as if he should feel disgraced because I voiced an opinion that wasn't popular. "Oh, you can't control your woman!"
The fact is, he doesn't own me, and as a free human being my thoughts can be expressed freely.
Walking on eggshells is a miserable way to live, especially for a married couple. When I speak, it's for the purpose of expressing what I believe. Not to mislead or manipulate, or to control a situation by hiding the truth.
Acknowledging a person's sovereignty over their own life is a happy way for adults to live together, because they respect each other. Not because one believes they own the other.