I haven't found the definition online or in one of my text books, so I would appreciate a confirmation that the following definition is correct (it's supposed to be a generalization of the twisted spin covariant derivative):
For all $i\in\{1,\ldots,n\}$, let $E_i\to M$ be a vector bundle with a covariant derivative $\nabla_i$ and consider the tensor product bundle $$E=E_1\otimes\cdots\otimes E_n.$$ The twisted covariant derivative \begin{equation*} \nabla\colon\Omega^0(M,E)\to\Omega^1(M,E) \end{equation*} is the unique linear function satisfying \begin{equation*} \nabla(s_1\otimes\cdots\otimes s_n)=\sum_{i=1}^ns_1\otimes\cdots\otimes s_{i-1}\otimes \nabla_is_i\otimes s_{i+1}\otimes\cdots\otimes s_n \end{equation*} for all sections $s_i\in\Gamma(M,E_i)$.