Let $x$ and $y$ be real numbers such that 
\[
  2 < \frac{x - y}{x + y} < 5.
\]
If $\frac{x}{y}$ is an integer, what is its value?
1. Let $\frac{x}{y} = m$, where $m$ is an integer. Then we can rewrite the given inequality in terms of $m$:
   \[
   2 < \frac{x - y}{x + y} < 5.
   \]
   Substituting $x = my$ into the inequality, we get:
   \[
   2 < \frac{my - y}{my + y} < 5.
   \]
2. Simplify the fraction:
   \[
   \frac{my - y}{my + y} = \frac{y(m - 1)}{y(m + 1)} = \frac{m - 1}{m + 1}.
   \]
   Thus, the inequality becomes:
   \[
   2 < \frac{m - 1}{m + 1} < 5.
   \]

3. Consider the case where $m + 1 > 0$:
   \[
   2 < \frac{m - 1}{m + 1} < 5.
   \]
   This implies:
   \[
   \frac{m - 1}{m + 1} < 1,
   \]
   which is a contradiction because $2 < 1$ is false. Therefore, $m + 1 \leq 0$, which means:
   \[
   m \leq -1.
   \]

4. Now, consider the inequality $2 < \frac{m - 1}{m + 1} < 5$ under the condition $m + 1 < 0$:
   \[
   2 < \frac{m - 1}{m + 1}.
   \]
   This can be rewritten as:
   \[
   2(m + 1) < m - 1,
   \]
   which simplifies to:
   \[
   2m + 2 < m - 1,
   \]
   \[
   2m - m < -1 - 2,
   \]
   \[
   m < -3.
   \]

5. Next, consider the other part of the inequality:
   \[
   \frac{m - 1}{m + 1} < 5.
   \]
   This can be rewritten as:
   \[
   m - 1 < 5(m + 1),
   \]
   which simplifies to:
   \[
   m - 1 < 5m + 5,
   \]
   \[
   -1 - 5 < 5m - m,
   \]
   \[
   -6 < 4m,
   \]
   \[
   -\frac{6}{4} < m,
   \]
   \[
   -\frac{3}{2} < m.
   \]

6. Combining the two inequalities:
   \[
   -3 < m < -\frac{3}{2}.
   \]
   Since $m$ is an integer, the only integer value that satisfies this inequality is:
   \[
   m = -2.
   \]

The final answer is $\boxed{-2}$.