A space-time physical-layer network coding (ST- PNC) method is presented for information exchange among multiple users over fully-connected multi-way relay networks. The method involves two steps: i) side-information learning and ii) space-time relay transmission. In the first step, different sets of users are scheduled to send signals over networks and the remaining users and relays overhear the transmitted signals, thereby learning the interference patterns. In the second step, multiple relays cooperatively send out linear combinations of signals received in the previous phase using space-time precoding so that all users efficiently exploit their side-information in the form of: 1) what they sent and 2) what they overheard in decoding. This coding concept is illustrated through two simple network examples. It is shown that ST-PNC improves the sum of degrees of freedom (sum-DoF) of the network compared to existing interference management methods. With ST-PNC, the sum-DoF of a general multi-way relay network without channel knowledge at the users is characterized in terms of relevant system parameters, chiefly the number of users, the number of relays, and the number of antennas at relays. A major implication of the derived results is that efficiently harnessing both transmit- ted and overheard signals as side-information brings significant performance improvements to fully-connected multi-way relay networks.