What is the difference between syntactic and semantic completeness?

It is apparent to me what the difference between syntactic consistency and semantic consistency is.

A theory $T$ is syntactically consistent if there exists no formula $\phi$ in the language such that both $\phi$ and $\neg \phi$ are provable.

A theory $T$ is semantically consistent if it has a model. If $T$ has a model, there exists an interpretation where all formulas of $T$ are true.

However, I do not understand the difference between syntactic completeness and semantic completeness. My understanding of the two properties is:

A theory $T$ is syntactically complete if for every formula $\phi$ in the language of the theory, either $\phi$ or $\neg \phi$ is provable.

A theory $T$ is semantically complete if, in every interpretation, every true formula $\phi$ is provable.

I do not understand how syntactic completeness can be false while semantic completeness can be true at the same time. I understand that this is true (it's true in any first order theory that is subject to Gödel's incompleteness theorem), I just do not see how they are not always true at the same time.

Take $T$ to be predicate logic with equality. Any sentence that is true in every model of $T$ is provable (by Gödel's completeness theorem), so $T$ is semantically complete. Now take $\phi$ to be $\forall x\cdot \forall y\cdot x = y$. Neither $\phi$ nor $\lnot\phi$ can be provable, because $\phi$ is true in some models but not in others. So $T$ is not syntactically complete.

The key is that syntactical consistency and completeness are defined relative to a proof system. That is, the notions of syntactical consistency and completeness refer to provability, and provability is always relative to a proof system.

Now, if you have a sound and complete proof system, then $T$ is syntactically complete if and only if $T$ is semantically complete, and the same goes for syntactic consistency vs semantic consistency.

However, if your proof system is unsound, or incomplete, then the two will diverge. Consider, for example, if your proof system contains the following inference rule:

$$\frac{}{\therefore P}\qquad \text{(Hokus Ponens)}$$

Well, such a proof system can of course prove everything, and any theory $T$ will be syntactically complete and syntactically inconsistent relative to this proof system, even as $T$ could well be semantically incomplete and semantically consistent.