This phenomenon is known as a chain-split (i.e. the bitcoin network can no longer agree which chain is the canonical chain).
Nodes will be network-partitioned until one chain becomes dominant.
Is the information of the length and accumulated difficulty broadcasted along with mined block?
How will the recipient node handle this situation?
Nodes will handle equal branches on a First Come, First Serve basis, so there will be a network partition where some nodes agree chain A with length N+4 is the canonical chain and the other set of nodes will assume chain B with length N+4 is the canonical chain. In the event of such a chain-split, the race is on for miners to produce a new block-tip and broadcast it across the bitcoin network ASAP to achieve convergence. When the contention is resolved, the node can switch to longer branch and obsolete the old branch.