CSP Forward checking with n-ary (and binary) constraints

I have implemented my own CSP solver using a Backtracking algorithm. Within the Backtracking algorithm I apply a Forward Checking algorithm (reducing domains of connected, unnasigned variables) that only works with Binary constraints.

The problem I have is that the CSP I try to solve has lots of n-ary constraints, making the FC algorithm mostly redundant.

Is there a way to enhance or replace my FC algorithm, so it works with n-ary constraints. Or is there a "simple" procedure to convert n-ary constraints to binary constraints?

EDIT: Added pseudo code (attempt) of the Forward Checking algorithm:

function ForwardChecking(variable, csp) list of variables with new domains, or failure
for each constraint in connectedConstraints(variable) do
if connectedVariable is assigned then
if constraint is not satisfied then
return failure
else
for each value in connectedVariable.domain do
connectedVariable.assign(value)
if constraint is not satisfied do
remove value from connectedVariable.domain
if connectedVariable.domain.count == 0 then
return failure