By H-space I mean a unital magma in $Ho(\mathsf{Top}_*)$, i.e. the homotopy category of pointed spaces (feel free to use any of your favourite model :) ).

This means that we have a pointed space $X$ and a map $\mu_X:X \times X \to X$ such that $\mu_X \circ j \simeq \nabla$, where $\nabla : X \vee X \to X$ is the fold (or co-diagonal) map, and $j:X \vee X \to X \times X$ is the pushout-product of the map $* \to X$ with itself.

An H-map between H-spaces is a map $f:X \to Y$ of the underlying spaces such that the obvious square involving $f$ and the multiplication maps is homotopy commutative.

Here comes the question: how to prove that the homotopy fibre of $f$ is again an H-space (in a sensible way, of course)?

Homotopy pullbacks of spaces are weak pullbacks in the homotopy category, and that's how you get a multiplication on the fibre compatible with that of $X$. However, I can't seem to find how to check the desired (homotopy) commutativity, as that would seem to require that weak pullback to be a strict one, which is known to be false.

Thanks in advance for any answer/comment!