  In 2005, Balmer defined the ringed space $\operatorname{Spec}_\otimes
\mathcal{T}$ for a given tensor triangulated category, while in 2023, the
second author introduced the ringed space $\operatorname{Spec}_\vartriangle
\mathcal{T}$ for a given triangulated category. In the algebro-geometric
context, these spectra provided several reconstruction theorems using derived
categories. In this paper, we prove that
$\operatorname{Spec}_{\otimes_X^\mathbb{L}} \operatorname{Perf} X$ is an open
ringed subspace of $\operatorname{Spec}_\vartriangle \operatorname{Perf} X$ for
a quasi-projective variety $X$. As an application, we provide a new proof of
the Bondal-Orlov and Ballard reconstruction theorems in terms of these spectra.
  Recently, the first author introduced the Fourier-Mukai locus
$\operatorname{Spec}^\mathsf{FM} \operatorname{Perf} X$ for a smooth projective
variety $X$, which is constructed by gluing Fourier-Mukai partners of $X$
inside $\operatorname{Spec}_\vartriangle \operatorname{Perf} X$. As another
application of our main theorem, we demonstrate that
$\operatorname{Spec}^\mathsf{FM} \operatorname{Perf} X$ can be viewed as an
open ringed subspace of $\operatorname{Spec}_\vartriangle \operatorname{Perf}
X$. As a result, we show that all the Fourier-Mukai partners of an abelian
variety $X$ can be reconstructed by topologically identifying the Fourier-Mukai
locus within $\operatorname{Spec}_\vartriangle \operatorname{Perf} X$.
