The dog bone space is given by a polyhedral decomposition \(\mathcal P\) of the plane \(\mathbb R^2\) given by two vertices \(v_1 = (0,0)\) and \(v_2 = (1,1)\), one-dimensional cells given by the line connecting \(v_1\) and \(v_2\) and then four unbounded edges extending downward and to the left connected to \(v_1\) and upward and to the right for \(v_2\), and then four two-dimensional cells.
To construct it, take this polyhedral decomposition, embed it as \(\mathbb R^2 = \mathbb R^2 \times \{1\} \subseteq \mathbb R^3\}\), and then take the cone over it. This produces a fan \(\Sigma\) whose support is \(\mathbb R^2\times\mathbb R_{\geq 0}\). Let \(X = X_{\Sigma}\).
The height projection on \(|\Sigma| \to \mathbb R_{\geq 0}\) produces a map \(\pi:X\to \mathbb A^1\). It’s generic section is isomorphic to \(\mathbb P^1\), so \(\pi^{-1}(\mathbb A^1\setminus \{0\}) = \mathbb P^1\times \mathbb P^1 \times (\mathbb A^1\setminus \{0\})\). It’s fiber over \(0\) is two copies of \(\mathbb P^2\)glued together along a toric strata, i.e. \(\pi^{-1}(0) = \mathbb P^2 \sqcup_{\mathbb P^1} \mathbb P^2\). See this on page 105, Example 5.5 of the Punctured Log Gromov-Witten Invariants paper.