The first step is to make your tests easier to review. One way to do it is by first writing high level scenarios without any test steps and getting one or two reliable people to review them. If the reviewers are of low quality, then you’ll end up wasting time and getting no results. Depending on how bad the situation is, you might have to do surprise checks initially to see if people are actually reviewing the tests or merely signing off on it.
E.g. Searching for products on eBay feature : (1) Product exists, (2) product does not exist, (3) Product is pending sale, (4) Product listing expired. Reviewer - You also need to add scenarios for products that are for adults only, available only in a certain region etc. Classify them as positive, negative etc.
The next steps is to write the test cases with steps for the scenarios. Depending on the time available, the reviewers can review all or some of the test cases. If a small sample is shoddy, then its likely that the rest is too.
If you use Jira or such, then add tasks for test reviews and assign it to people. Make sure that the feedback is given in the task itself vs in-person or in chats. You need to have some visibility into the review process. Remind your team that this is to help them test without confusion and not a tool for attacking each other.
PS - @sjprior please let me know if you feel there are any shortcomings in the above approach. I’d appreciate frank feedback. Thank you.