Limits of determinacy on reals For $X\subseteq\mathbb{R}^\omega$, say that $X$ is determined if the associated game on $\mathbb{R}$ of length $\omega$ (players I and II alternate playing reals, player I wins iff the sequence built is in $X$) is determined.
My question is: what is known about the consistency of determinacy principles for games played with reals? For example, is it consistent with large cardinals that every $X\subseteq\mathbb{R}^\omega$ is determined? (I suspect not, but I'm having trouble coming up with a counterexample.)

EDIT: This actually splits into two questions; I'm interested in each:

How much determinacy on $\mathbb{R}$ is consistent (relative to large cardinals) with ZF?

Andreas' answer completely settles this question.
However, I'm also interested in:

How much determinacy on $\mathbb{R}$ is consistent (relative to large cardinals) with ZFC?

 A: $\text{AD}_{\mathbb{R}}$ is equiconsistent with the existence of a $\lambda$ which is a limit of Woodin cardinals and cardinals which are $< \lambda$-strong. This large cardinal hypothesis is also known as the $\text{AD}_{\mathbb{R}}$-hypothesis. In other words this is also the same as having a strategic model with $\omega$ many Woodin cardinals.
A: The statement that every subset of $\mathbb R^\omega$ is determined is called $AD_{\mathbb R}$, and it's consistent relative to large cardinals.  I don't remember exactly how large, but I vaguely recall that it's only a little beyond what's needed for AD.  (Surely an expert will soon stop by and provide the exact answer.)
A: I know this is an old question, but here's a little information about the $\textsf{ZFC}$ case to anyone finding this question.
In my MO question, Juan kindly pointed out that Woodin has a result stating that it's consistent relative to a sharp for a Woodin limit of Woodins that every $\textsf{OD}(\mathbb R)$ game of length $\omega_1$ on the reals is determined. This result can be found in Neeman's book on long games, exercise 7F.15.
Beyond that, we can find a non-determined definable game of length $\omega_1+\omega$ on the reals (equivalently the integers). After $\omega_1$ many rounds, player I has played a sequence $X$ of reals. If $X$ contains a perfect subset then we can define a well-ordering of the reals via $X$ and thus a non-determined set of reals $A\subseteq \omega^\omega$. Player I is then supposed to spend his last $\omega$ moves to land in $A$, making the game non-determined. If $X$ did not contain a perfect subset then the last $\omega$ rounds is spent playing the perfect set game on $X$, which is non-determined as $X$ doesn't have the perfect set property. It seems like this game is $\Delta^2_2$, so that's at least a lower definability bound for inconsistency. When we get to length $\mathfrak c+\omega$ then we only have to consider the first case above, so that the game is (at most) $\Pi^1_2$ (for all strategies there's a move which makes the strategy non-winning).
I'm not sure what happens below these lower bounds, except that Borel determinacy at least ensures that all Borel games are determined, no matter the length (just play elements of ${^\alpha}\omega$ for $\omega$ many rounds instead of playing elements of $\omega$ for $\alpha\omega$ many rounds). Here's an attempt to illustrate the situation, where red indicates inconsistencies and blue indicates consistency relative to large cardinals. I'm not sure if more is known.

