Let $M$ be a connected projective complex manifold with a smooth anticanonical divisor $D$ ($D \sim -K_M$). In an answer to a previous question, It is told that $D$ may have at most two components.

An easy example is $M= \mathbb P^1 \times X$, where $X$ is a projective manifold with trivial canonical class.

Any blow-up of $M$ along a smooth variety that is a smooth divisor of $D$ is again such an example.

Here is my question:

Is a projective manifold whose anticanonical section is composed of two components always birational to $\mathbb P^1 \times X$, where $X$ is a projective manifold with trivial canonical class?

Of course, it is true for two dimension. I am curious about higher dimensional cases.