The tabloid targets have welcomed VH1's cameras into their Los Angeles house to shoot LeAnn & Eddie, giving the duo a chance to dispel misconceptions about their romance. The country singer and the actor (who most recently starred on TV Land's Hot in Cleveland) married in 2011.
"People are always wanting to make it a story of me being a homewrecker and trying to take someone's kids," Rimes, 31, tells PEOPLE. "They take the love I have for my family and twist it around in some weird, sick way."
Although their series won't feature Cibrian's kids – Mason, 11, and Jake, 7 – from his previous marriage to reality star Brandi Glanville, he says it will show how fiercely devoted he is to his singer wife.
"The biggest global misconception would be that we're not two people who are really in love with each other, that there are ulterior motives to our relationship," says Cibrian, 41. "But people just don't get it. Hopefully people will get a chance to see how we just click."
For more on LeAnn & Eddie , which airs Thursdays (10:30 p.m. ET) on VH1, pick up the current issue of PEOPLE, on newsstands Friday