Are the components of a Bandpass signal namely "inphase" and "quadrature" real functions in time domain?
$X(t) = Re\{XX(t)e^{j2\pi ft}\}$. So $XX(t)$ can be any generic complex baseband signal with real I and Q components - $XX(t) = XX_i(t)+j XX_q(t)$ and $e^{j2\pi ft} = \cos(2\pi ft) + j\sin(2\pi ft)$

After the complex multiplication, and taking real part, you get $X(t) = XX_i(t)\cos(2\pi ft) - XX_q(t)\sin(2\pi ft)$.

So $XX_i(t)$ and $XX_q(t)$ are indeed real functions and their spectra is even-symmetric.

If $\hat{X}(t)$ is the Hilbert transform of $X(t)$, then $X^+(t) = X(t)+j\hat{X}(t)$. $XX_i(t)=X(t)\cos(2\pi ft)+\hat{X}(t)\sin(2\pi ft); XX_q(t)=\hat{X}(t)\cos(2\pi ft)-X(t)\sin(2\pi ft)$