Suppose $X$ and $Y$ are jointly distributed real-valued random variables and for all outcomes $\omega\_1$, $\omega\_2$, we have
$$
X(\omega\_1)\le X(\omega\_2)\quad\Longrightarrow\quad Y(\omega\_1)\le Y(\omega\_2).
$$


**Edit**: As Louigi Addario-Berry's answer below shows, it may be better to consider the following variation: 
$$
X(\omega\_1)< X(\omega\_2)\quad\Longrightarrow\quad Y(\omega\_1)\le Y(\omega\_2).
$$



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> Does this property have a name?
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Strongly correlated? Terminology question
This is along the lines of Tom's answer. $X$ induces a partial order on $\Omega$. In fact, it induces a total order on a partition of $\Omega$ into sets $X^{-1}(x)$, $x \in \mathbb{R}$); 
simply say $X^{-1}(x) < X^{-1}(y)$ if $x < y$. 


By your property, there is then some non-decreasing function $y:\mathbb{R} \to \mathbb{R}$ such that for $x \in \mathbb{R}$, if $\omega\_1, \omega\_2 \in X^{-1}(x)$ then $Y(\omega\_1)=Y(\omega\_2)=y(x)$. 


But then we can write $Y(\omega)=y(X(\omega))$. In other words, $Y$ is just a non-decreasing (measurable) function of $X$. 


