  A single Josephson junction in the phase-slip regime exhibits Bloch
oscillations in the voltage when biased with a DC current $I_\text{DC}$. The
frequency of the oscillation is given by $\pi I_\text{DC}/e$, with $e$ the
elementary charge, linking the current to the frequency via fundamental
constants of nature. If an additional AC drive is applied, the Bloch
oscillations may synchronize with the external drive. This leads to the
emergence of dual Shapiro steps at fixed current in the $IV$ characteristics of
the device. For applications as a current standard, frequencies of the order of
10\,GHz are required. These are challenging to implement experimentally without
detrimental effects due to stray capacitances. Here, we propose to employ an
additional Josephson junction with a DC voltage bias as an on chip AC source
due to the AC Josephson effect. We study the back action of the Bloch
oscillations on the Josephson oscillations and identify a parameter regime in
which it is minimized. Furthermore, we find that the back action can even be
utilized to further enhance the driving signal which can lead to increased
widths of the resulting dual Shapiro steps. Finally, we show dual Shapiro steps
for a set of realistic experimental parameters at finite temperatures.
