Remote clock synchronization is crucial for many classical and quantum network applications. Current state-of-the-art remote clock synchronization techniques achieve femtosecond-scale clock stability utilizing frequency combs, which are supplementary to quantum-networking hardware. Demonstrating an alternative, we synchronize two remote clocks across our freespace testbed using a method called two-way quantum time transfer (QTT). In one second we reach picosecond-scale timing precision under very lossy and noisy channel conditions representative of daytime space-Earth links with commercial off-the-shelf quantum-photon sources and detection equipment. This work demonstrates how QTT is potentially relevant for daytime space-Earth quantum networking and/or providing high-precision secure timing in GPS-denied environments.