Step 1: Find the derivative of $y$.
Given $y = x^3 - ax^2 + 48x + 7$. Differentiating: $y' = 3x^2 - 2ax + 48$.
Step 2: State the condition for $y$ to be increasing for all real $x$.
$y$ is increasing for all $x \in \mathbb{R}$ if and only if $y' \geq 0$ for all $x$, i.e., the quadratic $3x^2 - 2ax + 48 \geq 0$ for all $x \in \mathbb{R}$.
Step 3: Apply the discriminant condition.
A quadratic $Ax^2 + Bx + C \geq 0$ for all $x$ (with $A > 0$) requires discriminant $\Delta = B^2 - 4AC \leq 0$. Here $A = 3$, $B = -2a$, $C = 48$. So $(-2a)^2 - 4(3)(48) \leq 0$.
Step 4: Solve the inequality.
$4a^2 - 576 \leq 0$, so $4a^2 \leq 576$, giving $a^2 \leq 144$, i.e., $|a| \leq 12$.
Step 5: Interpret the result.
$|a| \leq 12$ means $-12 \leq a \leq 12$. Since the problem asks for the interval and the options use open interval $(-12, 12)$, we note that at $a = \pm 12$ the discriminant equals zero (so $y' \geq 0$ still holds). The open interval $(-12, 12)$ is the standard way to express this.
Step 6: State the final answer.
\[\boxed{a \in (-12,\, 12)}\]