Are there any chemicals with this structural skeleton?
CCDC doesn't contain any molecules of such geometry. The closest topology observed is among pnictogens such as $\ce{P, As, Sb}$, which form tetrahedron units $[\ce{P4}]$, $[\ce{As4}]$, $[\ce{Sb4}]$ which are readily coordinating with Au, Rh, Ag and Cu by edge-sharing: 



The first publication (1) tetraphosphido-ligand presents the following structure of $\ce{Rh}$-complex: 



The most relevant structure best fitting the required connectivity matrix would probably be bis(tricyclo[$1.1.0.0^{2,4}$]tetraphosphane)-gold (2):





Thanks to @TAR86 comment mentioning exotic boranes, I also found a research (3) on ab initio calculations for $\ce{B4}$ cores obtained via the following proposed reaction:



They also obtained crystal structure for bis(tetrahydropyran)-(tetra(t-butyl)tetraborane)-lithium where $\ce{B4H}$ core is present:



(1) Lindsell, W. E.; McCullough, K. J.; Welch, A. J. J. Am. Chem. Soc. 1983, 105 (13), 4487–4489. DOI: 10.1021/ja00351a067 
(2) Forfar, L. C.; Clark, T. J.; Green, M.; Mansell, S. M.; Russell, C. A.; Sanguramath, R. A.; Slattery, J. M. Chem. Commun. 2012, 48 (14), 1970–1972. DOI: 10.1039/C2CC15291F 
(3) Neu, A.; Mennekes, T.; Englert, U.; Paetzold, P.; Hofmann, M.; von Ragué Schleyer, P. Angew. Chem. Int. Ed. Engl. 1997, 36 (19), 2117–2119. DOI: 10.1002/anie.199721171