  Interferometric measurements with arrays of radio antennas are a powerful and
widely used technique in astronomy. Recently, this technique has been revisited
for the reconstruction of extensive air showers [1]. This radio-interferometric
technique exploits the coherence in the radio emission emitted by billions of
secondary shower particles to reconstruct the shower parameters, in particular
the shower axis and depth of the shower maximum $X_\mathrm{max}$. The accuracy
previously demonstrated on simulations with an idealized detector is very
promising. In this article we evaluate the potential of interferometric
$X_\mathrm{max}$ measurements using (simulated) inclined air showers with
sparse antenna arrays under realistic conditions. To determine prerequisites
for the application of the radio-interferometric technique with various antenna
arrays, the influence of inaccuracies in the time synchronisation between
antennas and its inter-dependency with the antenna density is investigated in
detail. We find a strong correlation between the antenna multiplicity (per
event) and the maximum acceptable time jitter, i.e., inaccuracy in the time
synchronisation. For data recorded with a time synchronisation accurate to
within 1 ns in the commonly used frequency band of 30 to 80 MHz, an antenna
multiplicity of $> 50$ is needed to achieve an $X_\mathrm{max}$ resolution of
$\sigma_{X_\mathrm{max}} \lesssim 20$ g cm$^{-2}$. For data recorded with 2 ns
accuracy, already $\gtrsim 200$ antennas are needed to achieve this
$X_\mathrm{max}$ resolution. Furthermore, we find no advantage reconstructing
$X_\mathrm{max}$ from data simulated at higher observation frequencies, i.e.,
up to several hundred MHz. Finally, we provide a generalisation of our results
from very inclined air showers to vertical geometries.
