verbs - Omitting "that is/that are" and "its/their"

The two works are similar and it is not just because they are both
from franchises (that are) notorious for (their) poorly-written
characters.

Is it acceptable to omit the "that are" in the above sentence, and if so, what is doing so called?

Similarly, is it acceptable to omit the "their", and if so, what is doing so called?

My gut tells me yes to both, because it "sounds fine." However, why then does omitting those words sound so wrong in the following examples?

The two works are similar and it is not just because they are both
from franchises (that are) notorious.

The two franchises are notorious for (their) badness.

Answer

The two works are similar and it is not just because they are both from franchises (that are) notorious for (their) poorly-written characters.

It is perfectly acceptable to omit the "that are" and "their" in your first sentence, as they are attached to relative clauses, and match the case when their omission is optional.

The two works are similar and it is not just because they are both from franchises (that are) notorious.

The use of "notorious" as a postpositive adjective in this case does indeed sound unnatural.

The two franchises are notorious for (their) badness.

It is perfectly fine to omit the "their" in the second sentence, depending on the importance of specificity of the "badness."