Audio phase shift using Hartley Transform
I think your notation is a bit fishy but I'll use it. Assuming that your $\text{phase}(k)$ calculation is correct, do a complex multiplication of $\text{real}(k) + i\ \text{imag}(k)$ by $e^{i\alpha} = \cos(alpha) + i\sin(alpha)$. That can be done using four real multiplications and a few add/sub. Then you need to reverse the process of how you obtained $\text{real}(k)$ and $\text{imag}(k)$:

$$x(k) = \text{real}(k) + \text{imag}(k)$$
$$x(-k) = \text{real}(k) - \text{imag}(k)$$

Finally retransform.