A compound subject is two or more individual noun phrases coordinated to form a single, longer noun phrase. Compound subjects cause many difficulties in the proper usage of grammatical agreement between the subject and other entities [verbs, pronouns, etc.]. In reality, these issues are not specific to compound subjects as such, coming up equally as well with compound noun phrases of all sorts, but the problems are most acute with compound subjects because of the large number of types of agreement occurring with such subjects.
As shown in the examples, for English compound subjects joined by and, the agreement rules are generally unambiguous, but sometimes tricky. For example, the compound subject you and I is treated equivalently to we, taking appropriate pronominal agreement ["our car", not "your car", "their car", etc.]. In languages with more extensive subject-verb agreement [e.g. Spanish or Arabic], the verb agreement is clearly revealed as also being first-person plural.
Compound subjects joined by and generally take a plural verb. However, there are exceptions. When compound subjects are thought of as a single unit, a singular verb is used e.g. Peanut butter and jelly is available in the cafeteria.[1]
For the subjects joined by or, however, the rules are often ill-defined, especially when two elements that differ in grammatical gender or grammatical number are coordinate. [The tendency, in such cases, is to rewrite the sentences to avoid the conjunction: e.g. "Sylvia and I each have our own car, and one of us is planning to sell their car". Note that this still has a compound subject using and as the conjunction, and uses "semi-informal" "generic their" to get around the "his or her" problem. This could be avoided with a further rewrite: "Either Sylvia will sell her car, or I will sell mine."