The phrase "serious disagreement" means a strong and important difference of opinion or argument between people that has significant consequences or impact. Full definition
The issues are ones of serious disagreement over the vision of education reform. (mooneyinstitute.org)
It's not easy to be with people in the midst of intense feelings, uncomfortable situations, and serious disagreements about moral and ethical decisions. (religion-online.org)
Some problems might be simple differences of opinion, others may be much more serious disagreements over money, children, relatives, or a thousand other issues of daily life. (aspenpsychologyandcounseling.com)