We compute the covariance of the galaxy power spectrum multipoles in perturbation theory, including the effects of nonlinear evolution, nonlinear and nonlocal bias, radial redshift-space distortions, arbitrary survey window, and shot noise. We rewrite the power spectrum FKP estimator in terms of the usual windowed galaxy fluctuations and the fluctuations in the number of galaxies inside the survey volume. We show that this leads to a stronger supersample covariance than assumed in the literature and causes a substantial leakage of Gaussian information. We decompose the covariance matrix into several contributions that provide an insight into its behavior for different biased tracers. We show that for realistic surveys, the covariance of power spectrum multipoles is already dominated by shot noise and super survey mode coupling in the weakly nonlinear regime. Both these effects can be accurately modeled analytically, making a perturbative treatment of the covariance very compelling. Our method allows for the covariance to be varied as a function of cosmology and bias parameters very efficiently, with survey geometry entering as fixed kernels that can be computed separately using fast fourier transforms (FFTs). We find excellent agreement between our analytic covariance and that estimated from BOSS DR12 Patchy mock catalogs in the whole range we tested, up to k =0.6 h /Mpc . This bodes well for application to future surveys such as DESI and Euclid.