Consider the following equivalence relation on $\{0,1\}^\omega$:

$x\simeq y$ iff there is $n\in\omega$ such that $x(k)=y(k)$ for all $k\in\omega$ with $k\geq n$.

It is easy to see that the following is a well-defined ordering relation on $\{0,1\}^\omega/\simeq$ :

$[x]_\simeq \leq [y]_\simeq$ iff there is $n\in \omega$ such that $x(k)\leq y(k)$ for all $k\in\omega$ with $k\geq n$.

Let us order $\{0,1\}^\omega$ componentwise. Is there an order-preserving surjective function $f:(\{0,1\}^\omega/\simeq)\to \{0,1\}^\omega$?