Partial matrices whose completions all have the same rank

A partial matrix over a field F is a matrix whose entries are either elements of F or independent indeterminates. A completion of such a partial matrix is obtained by specifying values from F for the indeterminates. We determine the maximum possible number of indeterminates in an m × n partial matrix (m⩽n) whose completions all have a particular rank r, and we fully describe those examples in which this maximum is attained, without any restriction on the field F.