Spectral Floer theory and tangential structures

In \cite{PS}, for a stably framed Liouville manifold $X$ we defined a Donaldson-Fukaya category $\mathcal{F}(X;\mathbb{S})$ over the sphere spectrum, and developed an obstruction theory for lifting quasi-isomorphisms from $\mathcal{F}(X;\mathbb{Z})$ to $\mathcal{F}(X;\mathbb{S})$. Here, we define a spectral Donaldson-Fukaya category for any `graded tangential pair' $\Theta \to \Phi$ of spaces living over $BO \to BU$, whose objects are Lagrangians $L\to X$ for which the classifying maps of their tangent bundles lift to $\Theta \to \Phi$. The previous case corresponded to $\Theta = \Phi = \{\mathrm{pt}\}$. We extend our obstruction theory to this setting. The flexibility to `tune' the choice of $\Theta$ and $\Phi$ increases the range of cases in which one can kill the obstructions, with applications to bordism classes of Lagrangian embeddings in the corresponding bordism theory $\Omega^{(\Theta,\Phi),\circ}_*$. We include a self-contained discussion of when (exact) spectral Floer theory over a ring spectrum $R$ should exist, which may be of independent interest.