  In formulating covariant closed string field theories, we have always used
closed string fields with the level-matching condition. Recently, open
superstring field theories including the Ramond sector were constructed, and
one approach was to use open superstring fields with a constraint analogous to
the level-matching condition for closed string fields. There is another
approach developed by Sen, where no constraints are imposed on open superstring
fields but additional free string fields are introduced so that covariant
kinetic terms for the Ramond sector were constructed. Motivated by this
development, we construct closed string field theory without imposing the
level-matching condition on the closed string field, while we introduce an
additional free string field. This is the first implementation of general
covariance in the context of string theory without using the level-matching
condition. We explicitly expand the string fields up to the level relevant to
massless fields, and we confirm the equivalence to the theory with the
level-matching condition up to decoupled free fields.
