  Context. Solar Orbiter and PSP jointly observed the solar wind for the first
time in June 2020, capturing data from very different solar wind streams, calm
and Alfv\'enic wind as well as many dynamic structures. Aims. The aim here is
to understand the origin and characteristics of the highly dynamic solar wind
observed by the two probes, in particular in the vicinity of the heliospheric
current sheet (HCS). Methods. We analyse the plasma data obtained by PSP and
Solar Orbiter in situ during the month of June 2020. We use the Alfv\'en-wave
turbulence MHD solar wind model WindPredict-AW, and perform two 3D simulations
based on ADAPT solar magnetograms for this period. Results. We show that the
dynamic regions measured by both spacecraft are pervaded with flux ropes close
to the HCS. These flux ropes are also present in the simulations, forming at
the tip of helmet streamers, i.e. at the base of the heliospheric current
sheet. The formation mechanism involves a pressure driven instability followed
by a fast tearing reconnection process, consistent with the picture of
R\'eville et al. (2020a). We further characterize the 3D spatial structure of
helmet streamer born flux ropes, which seems, in the simulations, to be related
to the network of quasi-separatrices.
