If $L_1$ and $L_2$ are both non-decidable languages (Not decidable, so can be SD or $\lnot$SD), is it possible that $L_1-L_2$ is regular and $L_1-L_2\neq\phi$, where $\phi$ is the empty set?

What's a good way of tackling this question? Me and some classmates have been stuck on this question for a while, and this was the best we've got:

It should be possible given the following example:

Let $L_1=H\cup a^*$ and $L_2=H$, where $H=\{<M,w>:$ Turing machine $M$ halts on $w$$\}$, so $L_1-L_2=L_1\cap \lnot L_2=(H\cup a^*)\cap \lnot H=\lnot H\cap a^* = a^*$ which is regular. (we are saying $H$ does not contain $a^*$, so $\lnot H$ must contain $a^*$, hence why the intersection gives $a^*$.)

But we are not confident this is correct due to the last part with the intersection does not feel right. What's the answer to this question?