  Blowup equations and holomorphic anomaly equations are two universal yet
completely different approaches to solve refined topological string theory on
local Calabi-Yau threefolds corresponding to A- and B-model respectively. The
former originated from comparing Nekrasov partition functions of 4d
$\mathcal{N}=2$ gauge theories on $\Omega$ defomed spacetime
$\mathbb{C}^2_{\epsilon_1,\epsilon_2}$ and its one-point blown-up, while the
latter takes root in the degeneration of wordsheet Riemann surfaces. The
relation between the two approaches is an open question. In this short note, we
find a novel recursive equation governing their consistency, which we call the
consistency equation. This new equation computes the modular anomaly of blowup
equations order by order. The consistency equation also suggests a
non-holomorphic extension of blowup equations.
