Replica Identity

TL;DR: You need to specify which old values are required.

You may receive warning, that some field is not in WAL and search replica works in upsert only mode.
This is because old values for PK or Routing field is not available in updates stream.

If you define Routing field or use PK which is not Postgres PRIMARY KEY, you may need to configure it.

-- E.G:
ALTER TABLE mytable REPLICA IDENTITY FULL;

According to docs there four modes:

DEFAULT Records the old values of the columns of the primary key, if any.
USING INDEX index_name Records the old values of the columns covered by the named uniq index
FULL Records the old values of all columns in the row.
NOTHING Records no information about the old row.

Using index

Index must be unique, not partial, not deferrable, and include only columns marked NOT NULL.
Columns included via ... INCLUDE ( col1, col2, would be ignored by REPLICA IDENTITY.

CREATE UNIQUE INDEX example_uniq_idx ON mytable (
    foo,
    bar
    -- ALL YOUR FIELDS SHOULD BE HERE
) INCLUDE (
    baz
    -- THEESE FIELDS WOULD BE IGNORED
);
ALTER TABLE mytable REPLICA IDENTITY USING INDEX example_uniq_idx;