Here are a few examples:
$$(-1)^{\sqrt{2}},(-2)^{\pi},(-3)^{e}$$
From what I've learned, negative bases must have denominators of the exponent odd.
Normally if we do $(-2)^{0.258}$ it would be the same as $(-2)^{50/129}$. That means that since $129$ is odd, it will give a real number.
But for irrational numbers I'm not sure if we can determine if it's going to be odd on the denominator or not because you can't convert it to a fraction.
Hint: Express the numbers as such $$-1=e^{i\pi},\quad-2=2\cdot e^{i\pi}$$ and see if you can go from there.
By the way, it is not true that if the denominator is odd, the result will be real for you must bear in mind that there are always complex roots lurking in the background.
E.g. $\sqrt[3]{1}= 1, e^{\frac{i2\pi}{3}},e^{\frac{i4\pi}{3}}$