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It is consistent that there exists a non-principal ultrafilter over $\omega$ while $\mathbb{R}$ is not well-ordered. To see this, suppose that the partition relation $\omega \to (\omega)^{\omega}$ holds in $L(\mathbb{R})$. Then forcing with $\mathbb{P}= [\omega]^{\omega}$ adjoins a selective ultrafilter $\mathcal{U}$ over $\omega$ and $\mathcal{P}(\omega)$ cannot be well-ordered in $L(\mathbb{R})[\mathcal{U}]$. (See Eisworth's paper: Selective ultrafilters and $\omega \to (\omega)^{\omega}$.) Thus $L(\mathbb{R})[\mathcal{U}]$ is a model of $ZF$ which contains the nonprincipal ultrafilter $\mathcal{U}$ and yet $\mathbb{R}$ cannot be well-ordered in $L(\mathbb{R})[\mathcal{U}]$.

An update: it is perhaps also interesting to note that in $L(\mathbb{R})[\mathcal{U}]$, the ultraproduct $\prod_{\mathcal{U}} \bar{\mathbb{F}}_{p}$ of the algebraic closures of the fields of prime order $p$ is not isomorphic to $\mathbb{C}$.

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It is consistent that there exists a non-principal ultrafilter over $\omega$ while $\mathbb{R}$ is not well-ordered. To see this, suppose that the partition relation $\omega \to (\omega)^{\omega}$ holds in $L(\mathbb{R})$. Then forcing with $\mathbb{P}= [\omega]^{\omega}$ adjoins a selective ultrafilter $\mathcal{U}$ over $\omega$ and $\mathcal{P}(\omega)$ cannot be well-ordered in $L(\mathbb{R})[\mathcal{U}]$. (See Eisworth's paper": Selective ultrafilters and $\omega \to (\omega)^{\omega}$.) Thus $L(\mathbb{R})[\mathcal{U}]$ is a model of $ZF$ which contains the nonprincipal ultrafilter $\mathcal{U}$ and yet $\mathbb{R}$ cannot be well-ordered in $L(\mathbb{R})[\mathcal{U}]$.

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It is consistent that there exists a non-principal ultrafilter over $\omega$ while $\mathbb{R}$ is not well-ordered. To see this, suppose that the partition relation $\omega \to (\omega)^{\omega}$ holds in $L(\mathbb{R})$ and force with $\mathbb{P} = [\omega]^{\omega}$ ordered by inclusion. L(\mathbb{R})$. Then forcing with$\mathbb{P}$\mathbb{P}= [\omega]^{\omega}$ adjoins a selective ultrafilter $\mathcal{U}$ over $\omega$ and $\mathcal{P}(\omega)$ cannot be well-ordered in $L(\mathbb{R})[\mathcal{U}]$. (See Eisworth's paper "Selective ultrafilters and $\omega \to (\omega)^{\omega}$.) Thus $L(\mathbb{R})[\mathcal{U}]$ is a model of $ZF$ which contains the nonprincipal ultrafilter $\mathcal{U}$ and yet $\mathbb{R}$ cannot be well-ordered in $L(\mathbb{R})[\mathcal{U}]$.

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