  We present an approximately $C^1$-smooth multi-patch spline construction
which can be used in isogeometric analysis (IGA). A key property of IGA is that
it is simple to achieve high order smoothness within a single patch. To
represent more complex geometries one often uses a multi-patch construction. In
this case, the global continuity for the basis functions is in general only
$C^0$. Therefore, to obtain $C^1$-smooth isogeometric functions, a special
construction for the basis is needed. Such spaces are of interest when solving
numerically fourth-order problems, such as the biharmonic equation or
Kirchhoff-Love plate/shell formulations, using an isogeometric Galerkin method.
  Isogeometric spaces that are globally $C^1$ over multi-patch domains can be
constructed as in (Collin, Sangalli, Takacs; CAGD, 2016) and (Kapl, Sangalli,
Takacs; CAGD, 2019). The constructions require so-called analysis-suitable
$G^1$ parametrizations. To allow $C^1$ spaces over more general multi-patch
parametrizations, we need to increase the polynomial degree and relax the $C^1$
conditions. We adopt the approximate $C^1$ construction for two-patch domains,
as developed in (Weinm\"uller, Takacs; CMAME, 2021), and extend it to more
general multi-patch domains.
  We employ the construction for a biharmonic model problem and compare the
results with Nitsche's method. We compare both methods over complex multi-patch
domains with non-trivial interfaces. The numerical tests indicate that the
proposed construction converges optimally under $h$-refinement, comparable to
the solution using Nitsche's method. In contrast to weakly imposing coupling
conditions, the approximate $C^1$ construction is explicit and no additional
terms need to be introduced to stabilize the method. Thus, the new proposed
method can be used more easily as no parameters need to be estimated.
