Context. Solar Orbiter and Parker Solar Probe jointly observed the solar wind for the first time in June 2020, capturing data from very di ff erent solar wind streams: calm, Alfvénic wind and also highly dynamic large-scale structures. Context. Our aim is to understand the origin and characteristics of the highly dynamic solar wind observed by the two probes, particularly in the vicinity of the heliospheric current sheet (HCS). Methods. We analyzed the plasma data obtained by Parker Solar Probe and Solar Orbiter in situ during the month of June 2020. We used the Alfvén-wave turbulence magnetohydrodynamic solar wind model WindPredict-AW and we performed two 3D simulations based on ADAPT solar magnetograms for this period. Results. We show that the dynamic regions measured by both spacecraft are pervaded by flux ropes close to the HCS. These flux ropes are also present in the simulations, forming at the tip of helmet streamers, that is, at the base of the heliospheric current sheet. The formation mechanism involves a pressure-driven instability followed by a fast tearing reconnection process. We further characterize the 3D spatial structure of helmet streamer born flux ropes, which appears in the simulations to be related to the network of quasi-separatrices.