A more thorough explanation:

A back-in right is a type of reversionary interest in an oil-and-gas lease. It entitles the assignor to a share of the working interest after the assignee has recovered specified costs from production.

For example, let's say Company A assigns an oil-and-gas lease to Company B. The lease includes a back-in right for Company A, which means that once Company B recovers its specified costs from production, Company A has the right to regain a share of the working interest in the lease.

Another example could be a landowner who leases their land to an oil company. The lease agreement includes a back-in right for the landowner, which allows them to regain a share of the working interest in the lease after the oil company recovers its specified costs from production.

Overall, a back-in right is a way for the assignor to retain some ownership and control over the oil-and-gas lease, even after assigning it to someone else.