  In this paper, we consider a soliton gas of the focusing modified Korteweg-de
Vries generated from the $N$-soliton solutions under the nonzero background.
The spectral soliton density is chosen on the pure imaginary axis, excluding
the branch cut $\Sigma_{c}=\left[-i, i\right]$. In the limit $N\to\infty$, we
establish the Riemann-Hilbert problem of the soliton gas. Using the Deift-Zhou
nonlinear steepest-descent method, this soliton gas under the nonzero
background will decay to a constant background as $x\to+\infty$, while its
asymptotics as $x\to-\infty$ can be expressed with a Riemann-Theta function,
attached to a Riemann surface with genus-two. We also analyze the large $t$
asymptotics over the entire spatial domain, which is divided into three
distinct asymptotic regions depending on the ratio $\xi=\frac{x}{t}$. Using the
similar method, we provide the leading-order asymptotic behaviors for these
three regions and exhibit the dynamics of large $t$ asymptotics.
