  We generalize the notion of spectral triple with reality structure to
spectral triples with multitwisted real structure, the class of which is closed
under the tensor product composition. In particular, we introduce a
multitwisted order one condition (characterizing the Dirac operators as an
analogue of first-order differential operator). This provides a unified
description of the known examples, which include rescaled triples with the
conformal factor from the commutant of the algebra and (on the algebraic level)
triples on quantum disc and on quantum cone, that satisfy twisted first order
condition of \cite{BCDS16,BDS19}, as well as asymmetric tori, non-scalar
conformal rescaling and noncommutative circle bundles. In order to deal with
them we allow twists that do not implement automorphisms of the algebra of
spectral triple.
