Find all polynomials $W$ with real coefficients possessing the following property: if $x+y$ is a rational number, then $W(x)+W(y)$ is rational.
1. **Initial Setup and Assumptions:**
   We are given a polynomial \( W \) with real coefficients such that if \( x + y \) is rational, then \( W(x) + W(y) \) is rational. We need to find all such polynomials.

2. **Analyzing the Condition:**
   Fix a rational number \( q \). Consider the function \( f(x) = W(x) + W(q - x) \). Since \( x + (q - x) = q \) is rational, \( f(x) \) must be rational for all \( x \). 

3. **Continuity and Constancy:**
   The function \( f(x) \) is continuous (since \( W \) is a polynomial and hence continuous) and takes only rational values. A continuous function that takes only rational values must be constant. Therefore, \( f(x) = c \) for some constant \( c \).

4. **Expressing \( W \) in Terms of \( Z \):**
   We have \( W(x) + W(q - x) = c \). Let \( Z(x) = W(x) - W(0) \). Then:
   \[
   Z(x) + Z(q - x) = W(x) - W(0) + W(q - x) - W(0) = W(x) + W(q - x) - 2W(0) = c - 2W(0)
   \]
   Let \( k = c - 2W(0) \). Thus, \( Z(x) + Z(q - x) = k \).

5. **Additivity on Rationals:**
   If \( x + y \) is rational, then:
   \[
   W(x) + W(y) = W(x + y) + W(0)
   \]
   Substituting \( Z(x) = W(x) - W(0) \), we get:
   \[
   Z(x) + Z(y) = Z(x + y)
   \]
   This shows that \( Z \) is additive on the rationals.

6. **Induction on Rational Multiples:**
   By induction, we can show that \( Z(nx) = nZ(x) \) for any integer \( n \) and rational \( x \). For rational \( x = \frac{m}{n} \):
   \[
   nZ\left(\frac{m}{n}\right) = Z(m) = mZ(1)
   \]
   Therefore, \( Z\left(\frac{m}{n}\right) = \frac{m}{n}Z(1) \), implying \( Z(q) = qZ(1) \) for all rational \( q \).

7. **Form of \( W \):**
   Since \( Z(x) = xZ(1) \) for all rational \( x \), and \( Z(x) = W(x) - W(0) \), we have:
   \[
   W(x) = W(0) + x(W(1) - W(0))
   \]
   This is a linear polynomial of the form \( W(x) = ax + b \) where \( a = W(1) - W(0) \) and \( b = W(0) \).

8. **Rational Coefficients:**
   Since \( W(x) + W(y) \) must be rational whenever \( x + y \) is rational, \( a \) and \( b \) must be rational numbers.

Conclusion:
The only polynomials \( W \) with the given property are of the form \( W(x) = ax + b \) where \( a \) and \( b \) are rational numbers.

The final answer is \( \boxed{ W(x) = ax + b } \) where \( a \) and \( b \) are rational numbers.