I realized the answer is almost certainly "no", so I asked a better version of this question at How much of a variety can be reconstructed from codimension-zero data?.

I think that if you take two hypersurfaces in $\mathbb{P}^3$ of the same degree defined by two polynomials with mutually transcendental coefficients then they won't be distinguishable. The cohomology of the constant sheaf for a surface with punctures is concentrated in degrees $0,1$ and has only $\mathbb{C}$ in degree $0$, the DG algebra is uniquely determined by the vector space $H^1(O_X,X\setminus S)$ and therefore should boil down to some combinatorial data which generically won't depend on coefficients. (I'm not 100% sure of this since there may be some deformable data in the transition maps as you add points to $S$).

I realized the answer is almost certainly "no", so I asked a better version of this question at How much of a variety can be reconstructed from codimension-zero data?.

I think that if you take two hypersurfaces in $\mathbb{P}^3$ of the same degree defined by two polynomials with mutually transcendental coefficients then they won't be distinguishable. The cohomology of the constant sheaf for a surface with punctures is concentrated in degrees $0,1$ and has only $\mathbb{C}$ in degree $0$, the DG algebra is uniquely determined by the vector space $H^1(O_X,X\setminus S)$ and therefore should boil down to some combinatorial data which generically won't depend on coefficients. (I'm not 100% sure of this since there may be some deformable data in the transition maps as you add points to $S$).