Intermission: The dreaded parse and auxiliary verbs
When relative clauses come together in a sufficiently complex way, "parsing", or selecting the right relationship between words, gets increasingly complex. In all languages, the correct parse is selected with a combination of common sense and how common the possible patterns are.
Here's a popular meme about this exact phenomenon.
顔が赤い魚を食べた猫
(顔が赤い)(魚を食べた)猫
The cat that ate a fish and the cat had a red head.
((顔が赤い)魚)を食べた猫
The cat that ate a fish and the fish had a red head.
(顔が(赤い魚)を食べた)猫
The cat whose head ate a red fish.
顔が((赤い魚)を食べた猫)
The head is a cat that ate a red fish.
顔が赤い((魚を食べた)猫)
The head is a red cat that ate a fish.
Coming up in the next lesson, an auxiliary verb is a verb that attaches to another verb and gives meaning to that verb instead of having its own meaning.
Normal verbs like いる sometimes act as auxiliary verbs. Japanese japanese grammar distinguishes between auxiliary verbs that can either act normally or as auxiliaries, and auxiliary verbs that must only act as auxiliaries. So いる is sometimes called a "subsidiary" verb instead of auxiliary, even in english. In this guide we'll call them auxiliary when they're used that way, instead of treating "auxiliary verb" like a part of speech.