Learn Arabic (Modern Standard): هذا الاتفاق شبه رسمي.

شبه is technically a noun meaning "likeness" or "resemblance". To act as a prefix like "semi-" or "quasi-", it forms a mandatory Idafa (genitive construct) with the word that follows it. So, you are literally saying this agreement is "a resemblance of official".

What are the case endings on شبه رسمي if I read the sentence fully vocalized?

It would be pronounced شِبْهُ رَسْمِيٍّ (shibhu rasmiyyin). Because شبه is the predicate of the sentence (the core "news" about the subject), it takes the nominative case with a single damma. As the second part of the Idafa, رسمي must take the genitive case, shown here with kasratayn.

Since شبه رسمي is an Idafa, why is the second word رسمي indefinite (without ال)?

Because the entire phrase "semi-official" is the predicate of the sentence, and predicates are usually indefinite. If you added ال to make it هذا الاتفاق شبه الرسمي, you would be creating a definite descriptive phrase meaning "This semi-official agreement..." and the sentence would feel incomplete, as if it's waiting for a predicate.

As a quick refresher, how would the meaning change if I dropped the ال and wrote هذا اتفاق شبه رسمي?