To reuse an earlier comment, I don’t think normal people have strong enough commitments to meta-ethics either way. You will probably get different results from the general public depending on how you phrase the question, so if you pose the question like:

“Do you think different societies or individuals can have equally valid but conflicting moral views, with no single 'correct' answer?”

or

“Would you say that what is morally right or wrong depends on individual perspectives or societal norms, rather than being universal facts?”

You would get results that show the public supports moral anti-realism. However if you frame it like:

“Would you say certain moral principles (e.g. 'torturing babies is wrong') are true for everyone, everywhere, independent of personal opinions?”

You’d get results that show a favor for moral realism.

This either points to an internally contradictory belief system or an agnostic/underdeveloped belief system when it comes to meta-ethics.