What complexity class is this set of grammars? Given a grammar where every rule has the form $X \to YZ$, $XY \to Z$ or $X \to a$ where $X,Y,Z$ range over nonterminals and $a$ ranges over terminals, and given a nonterminal $S$ and a terminal $a$, determine whether $S$ can derive $a$.

What complexity class does this correspond to?

Related questions:

What complexity class does is this set of grammars? L-complete?

What complexity class does is this set of grammars? NL-complete?

What complexity class is this set of grammars? In between NL and P?

What complexity class is this set of grammars? RE?
Every unrestricted grammar can be converted to one with the following types of rules:
$$
AB \to CD \\
A \to BC \\
A \to a \\
A \to \epsilon
$$
Wikipedia calls this Kuroda normal form for unrestricted grammars.

You can convert a rule of the form $AB \to CD$ to a pair of rules $AB \to X_{ABCD}$, $X_{ABCD} \to CD$. Following the OP's suggestion, we can convert $A \to \epsilon$ to the rule $A \to E$, and add rules $BE \to B$ and $EB \to B$ for any nonterminal $B$. This allows us to obtain a grammar in your form equivalent to the original one, up to not being able to generate $\epsilon$.