Learn Arabic (Modern Standard): وجدت معطفا له جيب داخلي.

In Arabic, relative clauses that describe an indefinite noun like معطفا (a coat) do not use a relative pronoun. You simply place the descriptive clause له جيب داخلي right after the noun. You would only use الذي if the coat were definite, as in المعطف الذي له جيب داخلي.

How exactly does له جيب mean "that has a pocket"?

Arabic lacks a true verb for "to have" and uses prepositions instead. Here, له means "to it" (with the pronoun referring back to the coat). The phrase is literally "to it [is] a pocket," which is the standard Arabic way to express possession or having something.

If I pronounce the case endings, why is جيب (pocket) in the nominative case?

Because the phrase له جيب is a verbless nominal sentence, the word جيب acts as the delayed subject (the thing that exists "to the coat"). Subjects of nominal sentences always take the nominative case, even though "pocket" feels like an object in the English translation.

Why doesn't the adjective داخلي match the accusative case of معطفا?

The adjective داخلي (inner) describes the pocket, not the coat. Therefore, it must match the noun جيب. Since جيب is nominative as the subject of its own mini-sentence, داخلي must also be nominative.