= Solution
An <almost complex structure> $J$ on $(M,\omega_M)$ is a <compatible almost complex structure> when $J^2=-I$, $\omega_M(Ju,Jv)=\omega_M(u,v)$, and
$$
g_J(u,v)=\omega_M(u,Jv)
$$
is a positive-definite <inner product>. To prove existence, choose any <Riemannian metric> $h$ and define $A$ by $\omega_M(u,v)=h(Au,v)$. The <metric construction of a compatible almost complex structure>
$$
J=A(-A^2)^{-1/2}
$$
is smooth and compatible, so the space is nonempty.
Identify the compact symplectic manifold $N$ with its image under the <symplectic embedding>. Along $N$ there is a symplectic splitting
$$
TM|_N=TN\oplus(TN)^{\omega_M}.
$$
Choose compatible almost complex structures on both summands and take their direct sum. Its associated metric makes the two summands orthogonal. Extend this metric from the closed submanifold $N$ to all of $M$ using a <partition of unity>, and apply the metric construction again. Along $N$ it recovers the prescribed direct sum, so the resulting global compatible $J$ satisfies $J(TN)=TN$. This is the <relative extension of a compatible almost complex structure>.
For the two compact complex curves, use the supplied holomorphic coordinates at their transverse intersection. There $C_1\cup C_2$ is $\{xy=0\}$. Replace it in a small ball by the complex annulus $\{xy=\epsilon\}$ and use a cutoff in a surrounding annulus to rejoin the unchanged curves. For sufficiently small nonzero $\epsilon$, the result is an embedded symplectic surface $\Sigma$. This local replacement is a bordism between the old and new cycles, so
$$
[\Sigma]=[C_1]+[C_2]\in H_2(X;\mathbb Z).
$$
This is the <symplectic smoothing of a positive transverse node>.
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