Replacing $H$-space multiplication with a fibration

Question

Let $X$ be an $H$-space with multiplication $\mu: X \times X \to X$. Does there exist a space $\overset{\sim}{X}$ homotopy equivalent to $X$ such that the induced map $\overset{\sim}{\mu}: \overset{\sim}{X} \times \overset{\sim}{X} \to \overset{\sim}{X}$ is a fibration? Basic homotopy theory says that $\mu$ is equivalent to a fibration $(X \times X)' \to X'$, but I see no reason why such a replacement of $\mu$ would make $X'$ into an $H$-space.