  In strong gravitational lens systems, the light bending is usually dominated
by one main galaxy, but may be affected by other mass along the line of sight
(LOS). Shear and convergence can be used to approximate the contributions from
less significant perturbers (e.g. those that are projected far from the lens or
have a small mass), but higher order effects need to be included for objects
that are closer or more massive. We develop a framework for multiplane lensing
that can handle an arbitrary combination of tidal planes treated with shear and
convergence and planes treated exactly (i.e., including higher order terms).
This framework addresses all of the traditional lensing observables including
image positions, fluxes, and time delays to facilitate lens modelling that
includes the non-linear effects due to mass along the LOS. It balances accuracy
(accounting for higher-order terms when necessary) with efficiency (compressing
all other LOS effects into a set of matrices that can be calculated up front
and cached for lens modelling). We identify a generalized multiplane mass sheet
degeneracy, in which the effective shear and convergence are sums over the
lensing planes with specific, redshift-dependent weighting factors.
