ag.algebraic geometry - Embeddings and triangulations of real analytic varieties

I do not know an answer to the embedability question, but the triangulation can be deduced from the existence of a (Thom-Mather) stratification by [Johnson "On the triangulation of stratified sets and singular varieties", Trans. Amer. Math. Soc. 275 (1983), no. 1, p. 333–343] or [Goresky "Triangulation of stratified objects", Proc. Amer. Math.
Soc. 72 (1978), no. 1, p. 193–200]; moreover it is known that analytic subvarieties of Euclidean space are Whitney stratified [Whitney "Local properties of analytic varieties", Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N. J., 1965, p. 205–244 & "Tangents to an analytic variety", Ann. of Math. (2) 81 (1965), p. 496–549].

Moreover in [Mather "Notes on topological stability", 1970, Harvard University & "Stratifications and mappings", Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, New York, 1973, p. 195–232] you can find the result that Whitney stratified sets are Mather stratified. Finally, Mather stratification are given by local conditions (there might be an issue gluing the local strata but I do not think so), so these result should imply that analytic varieties are triangulable. I do not think it is the most efficient to do so.