Let $\mathbb{R}_*=\mathbb{R}^\omega/\mathcal U$ for some ultrafilter $\cal U$. In the definitions of this question and assuming ZFC + CH there are only three types of cuts in $\mathbb{R}_*$: $(\omega,\omega_1),~(\omega_1,\omega),~(\omega_1,\omega_1)$. And only $(\omega_1,\omega_1)$ cut could be filled. For an ordered field $\mathbb F$ lets say that a cut $\mathbf{F}=A\coprod B$ is good iff for any $c>0$ there are $a\in A$ and $b\in B$ such that $b-a<c$. Any good cut is of type $(\lambda,\lambda)$ where $\lambda$ is cofinality of $\mathbb F$. If any good cut is filled lets call the field quasi-complete. Any ordered field can be embedded into quasi-complete field with appropriate universal properties (see here).

Can we proof that (may be for special ultrafilter $\cal U$):

1. $\mathbb{R}_*$ is quasi-complete ?
2. If $\mathbb{F}$ is quasi-complete then $\mathbb{F}_*=\mathbb{F}^\omega/\mathcal{U}$ is quasi-complete ?
3. Any $(\omega_1,\omega_1)$ cut in $\mathbb{R}_*$ is good ?

Let $\mathbb{R}_*=\mathbb{R}^\omega/\mathcal U$ for some ultrafilter $\cal U$. In the definitions of this question and assuming ZFC + CH there are only three types of cuts in $\mathbb{R}_*$: $(\omega,\omega_1),~(\omega_1,\omega),~(\omega_1,\omega_1)$. And only $(\omega_1,\omega_1)$ cut could be filled. For an ordered field $\mathbb F$ lets say that a cut $\mathbb{F}=A\coprod B$ is good iff for any $c>0$ there are $a\in A$ and $b\in B$ such that $b-a<c$. Any good cut is of type $(\lambda,\lambda)$ where $\lambda$ is cofinality of $\mathbb F$. If any good cut is filled lets call the field quasi-complete. Any ordered field can be embedded into quasi-complete field with appropriate universal properties (see here).

Can we proof that (may be for special ultrafilter $\cal U$):

1. $\mathbb{R}_*$ is quasi-complete ?
2. If $\mathbb{F}$ is quasi-complete then $\mathbb{F}_*=\mathbb{F}^\omega/\mathcal{U}$ is quasi-complete ?
3. Any $(\omega_1,\omega_1)$ cut in $\mathbb{R}_*$ is good ?