My girlfriend often tells me that I'm sleeping with my eyes half-open. What could be the problem?

Nothing wrong. Many people sleep with their eyes partially open, without it being abnormal. Sometimes health issues such as thyroid can cause it. If eyes are irritated, ointment at bedtime to cover the exposed eyes will help.
PM lagophthalmos. Your description is likely normal in your case. It can also occur due to neurologic disease or after blepharoplasty lid surgery. If it bothers your eyes, you can try using ointment at night.