Cloud platforms commonly exploit workload temporal flexibility to reduce their carbon emissions. They suspend/resume workload execution for when and where the energy is greenest. However, increasingly prevalent delay-intolerant real-time workloads challenge this approach. To this end, we present a framework to harvest green renewable energy for real-time workloads in cloud systems. We use renewables-driven cores in servers to dynamically switch CPU cores between real-time and low-power profiles, matching renewable energy availability. We then develop a VM Execution Model to guarantee running VMs are allocated with cores in the real-time power profile. If such cores are insufficient, we conduct criticality-aware VM evictions as needed. Furthermore, we develop a VM Packing Algorithm to utilize available cores across the data center. We introduce the Green Cores concept in our algorithm to convert renewable energy usage into a server inventory attribute. Based on this, we jointly optimize for renewable energy utilization and reduction of VM eviction incidents. We implement a prototype of our framework in OpenStack as openstack-gc. Using an experimental openstack-gc cloud and a large-scale simulation testbed, we expose our framework to VMs running RTEval, a real-time evaluation program, and a 14-day Azure VM arrival trace. Our results show: (i) a 6.52% reduction in coefficient of variation of real-time latency over an existing workload temporal flexibility-based solution, and (ii) a joint 79.64% reduction in eviction incidents with a 34.83% increase in energy harvest over the state-of-the-art packing algorithms. We open source openstack-gc at https://github.com/tharindu-b-hewage/openstack-gc.