Phase-randomized optical homodyne detection is a well-known technique for performing quantum state tomography. So far, it has been mainly considered a sophisticated tool for laboratory experiments but unsuitable for practical applications. In this work, we change the perspective and employ this technique to set up a practical continuous-variable quantum random number generator. We exploit a phase-randomized local oscillator realized with a gain-switched laser to bound the min-entropy and extract true randomness from a completely uncharacterized input, potentially controlled by a malicious adversary. Our proof-of-principle implementation achieves an equivalent rate of 270 Mbit/s. In contrast to other source-device-independent quantum random number generators, the one presented herein does not require additional active optical components, thus representing a viable solution for future compact, modulator-free, certified generators of randomness.